Banach space

General theory

Linear operators, isomorphisms

If ${\displaystyle X}$ and ${\displaystyle Y}$ are normed spaces over the same ground field ${\displaystyle \mathbb{K},}$ the set of all continuous ${\displaystyle \mathbb{K}}$-linear maps ${\displaystyle T:X\to Y}$ is denoted by ${\displaystyle B(X,Y).}$ In infinite-dimensional spaces, not all linear maps are continuous. A linear mapping from a normed space ${\displaystyle X}$ to another normed space is continuous if and only if it is bounded on the closed unit ball of ${\displaystyle X.}$ Thus, the vector space ${\displaystyle B(X,Y)}$ can be given the operator norm $${\displaystyle \Vert T\Vert =sup\{\Vert Tx{\Vert }_Y\mid x\in X,\Vert x{\Vert }_X\le 1\}.}$$

For ${\displaystyle Y}$ a Banach space, the space ${\displaystyle B(X,Y)}$ is a Banach space with respect to this norm. In categorical contexts, it is sometimes convenient to restrict the function space between two Banach spaces to only the short maps; in that case the space ${\displaystyle B(X,Y)}$ reappears as a natural bifunctor.

If ${\displaystyle X}$ is a Banach space, the space ${\displaystyle B(X)=B(X,X)}$ forms a unital Banach algebra; the multiplication operation is given by the composition of linear maps.

If ${\displaystyle X}$ and ${\displaystyle Y}$ are normed spaces, they are isomorphic normed spaces if there exists a linear bijection ${\displaystyle T:X\to Y}$ such that ${\displaystyle T}$ and its inverse ${\displaystyle T^{-1}}$ are continuous. If one of the two spaces ${\displaystyle X}$ or ${\displaystyle Y}$ is complete (or reflexive, separable, etc.) then so is the other space. Two normed spaces ${\displaystyle X}$ and ${\displaystyle Y}$ are isometrically isomorphic if in addition, ${\displaystyle T}$ is an isometry, that is, ${\displaystyle \Vert T(x)\Vert =\Vert x\Vert }$ for every ${\displaystyle x}$ in ${\displaystyle X.}$ The Banach–Mazur distance ${\displaystyle d(X,Y)}$ between two isomorphic but not isometric spaces ${\displaystyle X}$ and ${\displaystyle Y}$ gives a measure of how much the two spaces ${\displaystyle X}$ and ${\displaystyle Y}$ differ.

Continuous and bounded linear functions and seminorms

Every continuous linear operator is a bounded linear operator and if dealing only with normed spaces then the converse is also true. That is, a linear operator between two normed spaces is bounded if and only if it is a continuous function. So in particular, because the scalar field (which is ${\displaystyle \mathbb{R}}$ or ${\displaystyle \mathbb{C}}$) is a normed space, a linear functional on a normed space is a bounded linear functional if and only if it is a continuous linear functional. This allows for continuity-related results (like those below) to be applied to Banach spaces. Although boundedness is the same as continuity for linear maps between normed spaces, the term "bounded" is more commonly used when dealing primarily with Banach spaces.

If ${\displaystyle f:X\to \mathbb{R}}$ is a subadditive function (such as a norm, a sublinear function, or real linear functional), then ${\displaystyle f}$ is continuous at the origin if and only if ${\displaystyle f}$ is uniformly continuous on all of ${\displaystyle X}$; and if in addition ${\displaystyle f(0)=0}$ then ${\displaystyle f}$ is continuous if and only if its absolute value ${\displaystyle |f|:X\to [0,\mathrm{\infty })}$ is continuous, which happens if and only if ${\displaystyle \{x\in X\mid |f(x)|<1\}}$ is an open subset of ${\displaystyle X.}$ And very importantly for applying the Hahn–Banach theorem, a linear functional ${\displaystyle f}$ is continuous if and only if this is true of its real part ${\displaystyle \mathrm{Re}f}$ and moreover, ${\displaystyle \Vert \mathrm{Re}f\Vert =\Vert f\Vert }$ and the real part ${\displaystyle \mathrm{Re}f}$ completely determines ${\displaystyle f,}$ which is why the Hahn–Banach theorem is often stated only for real linear functionals. Also, a linear functional ${\displaystyle f}$ on ${\displaystyle X}$ is continuous if and only if the seminorm ${\displaystyle |f|}$ is continuous, which happens if and only if there exists a continuous seminorm ${\displaystyle p:X\to \mathbb{R}}$ such that ${\displaystyle |f|\le p}$; this last statement involving the linear functional ${\displaystyle f}$ and seminorm ${\displaystyle p}$ is encountered in many versions of the Hahn–Banach theorem.

Basic notions

The Cartesian product ${\displaystyle X\times Y}$ of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are commonly used, such as $${\displaystyle \Vert (x,y){\Vert }_1=\Vert x\Vert +\Vert y\Vert ,\Vert (x,y){\Vert }_{\mathrm{\infty }}=max(\Vert x\Vert ,\Vert y\Vert )}$$ which correspond (respectively) to the coproduct and product in the category of Banach spaces and short maps (discussed above). For finite (co)products, these norms give rise to isomorphic normed spaces, and the product ${\displaystyle X\times Y}$ (or the direct sum ${\displaystyle X\oplus Y}$) is complete if and only if the two factors are complete.

If ${\displaystyle M}$ is a closed linear subspace of a normed space ${\displaystyle X,}$ there is a natural norm on the quotient space ${\displaystyle X/M,}$ $${\displaystyle \Vert x+M\Vert =\underset{m\in M}{inf}\Vert x+m\Vert .}$$

The quotient ${\displaystyle X/M}$ is a Banach space when ${\displaystyle X}$ is complete. The quotient map from ${\displaystyle X}$ onto ${\displaystyle X/M,}$ sending ${\displaystyle x\in X}$ to its class ${\displaystyle x+M,}$ is linear, onto, and of norm ${\displaystyle 1,}$ except when ${\displaystyle M=X,}$ in which case the quotient is the null space.

The closed linear subspace ${\displaystyle M}$ of ${\displaystyle X}$ is said to be a complemented subspace of ${\displaystyle X}$ if ${\displaystyle M}$ is the range of a surjective bounded linear projection ${\displaystyle P:X\to M.}$ In this case, the space ${\displaystyle X}$ is isomorphic to the direct sum of ${\displaystyle M}$ and ${\displaystyle \ker P,}$ the kernel of the projection ${\displaystyle P.}$

Suppose that ${\displaystyle X}$ and ${\displaystyle Y}$ are Banach spaces and that ${\displaystyle T\in B(X,Y).}$ There exists a canonical factorization of ${\displaystyle T}$ as $${\displaystyle T=T_1\circ \pi ,T:X\stackrel{\pi }{⟶}X/\ker T\stackrel{T_1}{⟶}Y}$$ where the first map ${\displaystyle \pi }$ is the quotient map, and the second map ${\displaystyle T_1}$ sends every class ${\displaystyle x+\ker T}$ in the quotient to the image ${\displaystyle T(x)}$ in ${\displaystyle Y.}$ This is well defined because all elements in the same class have the same image. The mapping ${\displaystyle T_1}$ is a linear bijection from ${\displaystyle X/\ker T}$ onto the range ${\displaystyle T(X),}$ whose inverse need not be bounded.

Classical spaces

Basic examples of Banach spaces include: the Lp spaces ${\displaystyle L^p}$ and their special cases, the sequence spaces ${\displaystyle {\ell }^p}$ that consist of scalar sequences indexed by natural numbers ${\displaystyle \mathbb{N}}$; among them, the space ${\displaystyle {\ell }^1}$ of absolutely summable sequences and the space ${\displaystyle {\ell }^2}$ of square summable sequences; the space ${\displaystyle c_0}$ of sequences tending to zero and the space ${\displaystyle {\ell }^{\mathrm{\infty }}}$ of bounded sequences; the space ${\displaystyle C(K)}$ of continuous scalar functions on a compact Hausdorff space ${\displaystyle K,}$ equipped with the max norm, $${\displaystyle \Vert f{\Vert }_{C(K)}=max\{|f(x)|\mid x\in K\},f\in C(K).}$$

According to the Banach–Mazur theorem, every Banach space is isometrically isomorphic to a subspace of some ${\displaystyle C(K).}$ For every separable Banach space ${\displaystyle X,}$ there is a closed subspace ${\displaystyle M}$ of ${\displaystyle {\ell }^1}$ such that ${\displaystyle X:={\ell }^1/M.}$

Any Hilbert space serves as an example of a Banach space. A Hilbert space ${\displaystyle H}$ on ${\displaystyle \mathbb{K}=\mathbb{R},\mathbb{C}}$ is complete for a norm of the form $${\displaystyle \Vert x{\Vert }_H=\sqrt{⟨x,x⟩},}$$ where $${\displaystyle ⟨\cdot ,\cdot ⟩:H\times H\to \mathbb{K}}$$ is the inner product, linear in its first argument that satisfies the following: $${\displaystyle \begin{array}{cc}⟨y,x⟩& =\overline{⟨x,y⟩},\text{ for all }x,y\in H\\ ⟨x,x⟩& \ge 0,\text{ for all }x\in H\\ ⟨x,x⟩=0\text{ if and only if }x& =0.\end{array}}$$

For example, the space ${\displaystyle L^2}$ is a Hilbert space.

The Hardy spaces, the Sobolev spaces are examples of Banach spaces that are related to ${\displaystyle L^p}$ spaces and have additional structure. They are important in different branches of analysis, Harmonic analysis and Partial differential equations among others.

Banach algebras

A Banach algebra is a Banach space ${\displaystyle A}$ over ${\displaystyle \mathbb{K}=\mathbb{R}}$ or ${\displaystyle \mathbb{C},}$ together with a structure of algebra over ${\displaystyle \mathbb{K}}$, such that the product map ${\displaystyle A\times A\ni (a,b)\mapsto ab\in A}$ is continuous. An equivalent norm on ${\displaystyle A}$ can be found so that ${\displaystyle \Vert ab\Vert \le \Vert a\Vert \Vert b\Vert }$ for all ${\displaystyle a,b\in A.}$

Examples

• The Banach space ${\displaystyle C(K)}$ with the pointwise product, is a Banach algebra.
• The disk algebra ${\displaystyle A(\mathbf{D})}$ consists of functions holomorphic in the open unit disk ${\displaystyle D\subseteq \mathbb{C}}$ and continuous on its closure: ${\displaystyle \overline{\mathbf{D}}.}$ Equipped with the max norm on ${\displaystyle \overline{\mathbf{D}},}$ the disk algebra ${\displaystyle A(\mathbf{D})}$ is a closed subalgebra of ${\displaystyle C\left(\overline{\mathbf{D}}\right).}$
• The Wiener algebra ${\displaystyle A(\mathbf{T})}$ is the algebra of functions on the unit circle ${\displaystyle \mathbf{T}}$ with absolutely convergent Fourier series. Via the map associating a function on ${\displaystyle \mathbf{T}}$ to the sequence of its Fourier coefficients, this algebra is isomorphic to the Banach algebra ${\displaystyle {\ell }^1(Z),}$ where the product is the convolution of sequences.
• For every Banach space ${\displaystyle X,}$ the space ${\displaystyle B(X)}$ of bounded linear operators on ${\displaystyle X,}$ with the composition of maps as product, is a Banach algebra.
• A C*-algebra is a complex Banach algebra ${\displaystyle A}$ with an antilinear involution ${\displaystyle a\mapsto a^{\ast }}$ such that ${\displaystyle \Vert a^{\ast }a\Vert =\Vert a{\Vert }^2.}$ The space ${\displaystyle B(H)}$ of bounded linear operators on a Hilbert space ${\displaystyle H}$ is a fundamental example of C*-algebra. The Gelfand–Naimark theorem states that every C*-algebra is isometrically isomorphic to a C*-subalgebra of some ${\displaystyle B(H).}$ The space ${\displaystyle C(K)}$ of complex continuous functions on a compact Hausdorff space ${\displaystyle K}$ is an example of commutative C*-algebra, where the involution associates to every function ${\displaystyle f}$ its complex conjugate ${\displaystyle \overline{f}.}$

Dual space

If ${\displaystyle X}$ is a normed space and ${\displaystyle \mathbb{K}}$ the underlying field (either the reals or the complex numbers), the continuous dual space is the space of continuous linear maps from ${\displaystyle X}$ into ${\displaystyle \mathbb{K},}$ or continuous linear functionals. The notation for the continuous dual is ${\displaystyle X^{\prime }=B(X,\mathbb{K})}$ in this article. Since ${\displaystyle \mathbb{K}}$ is a Banach space (using the absolute value as norm), the dual ${\displaystyle X^{\prime }}$ is a Banach space, for every normed space ${\displaystyle X.}$ The Dixmier–Ng theorem characterizes the dual spaces of Banach spaces.

The main tool for proving the existence of continuous linear functionals is the Hahn–Banach theorem.

Hahn–Banach theorem—Let ${\displaystyle X}$ be a vector space over the field ${\displaystyle \mathbb{K}=\mathbb{R},\mathbb{C}.}$ Let further

• ${\displaystyle Y\subseteq X}$ be a linear subspace,
• ${\displaystyle p:X\to \mathbb{R}}$ be a sublinear function and
• ${\displaystyle f:Y\to \mathbb{K}}$ be a linear functional so that ${\displaystyle \mathrm{Re}(f(y))\le p(y)}$ for all ${\displaystyle y\in Y.}$

Then, there exists a linear functional ${\displaystyle F:X\to \mathbb{K}}$ so that $${\displaystyle F{|}_Y=f,\text{ and }\text{ for all }x\in X,\mathrm{Re}(F(x))\le p(x).}$$

In particular, every continuous linear functional on a subspace of a normed space can be continuously extended to the whole space, without increasing the norm of the functional. An important special case is the following: for every vector ${\displaystyle x}$ in a normed space ${\displaystyle X,}$ there exists a continuous linear functional ${\displaystyle f}$ on ${\displaystyle X}$ such that $${\displaystyle f(x)=\Vert x{\Vert }_X,\Vert f{\Vert }_{X^{\prime }}\le 1.}$$

When ${\displaystyle x}$ is not equal to the ${\displaystyle \mathbf{0}}$ vector, the functional ${\displaystyle f}$ must have norm one, and is called a norming functional for ${\displaystyle x.}$

The Hahn–Banach separation theorem states that two disjoint non-empty convex sets in a real Banach space, one of them open, can be separated by a closed affine hyperplane. The open convex set lies strictly on one side of the hyperplane, the second convex set lies on the other side but may touch the hyperplane.

A subset ${\displaystyle S}$ in a Banach space ${\displaystyle X}$ is total if the linear span of ${\displaystyle S}$ is dense in ${\displaystyle X.}$ The subset ${\displaystyle S}$ is total in ${\displaystyle X}$ if and only if the only continuous linear functional that vanishes on ${\displaystyle S}$ is the ${\displaystyle \mathbf{0}}$ functional: this equivalence follows from the Hahn–Banach theorem.

If ${\displaystyle X}$ is the direct sum of two closed linear subspaces ${\displaystyle M}$ and ${\displaystyle N,}$ then the dual ${\displaystyle X^{\prime }}$ of ${\displaystyle X}$ is isomorphic to the direct sum of the duals of ${\displaystyle M}$ and ${\displaystyle N.}$ If ${\displaystyle M}$ is a closed linear subspace in ${\displaystyle X,}$ one can associate the orthogonal of ${\displaystyle M}$ in the dual, $${\displaystyle M^{\mathrm{\perp }}=\{x^{\prime }\in X\mid x^{\prime }(m)=0\text{ for all }m\in M\}.}$$

The orthogonal ${\displaystyle M^{\mathrm{\perp }}}$ is a closed linear subspace of the dual. The dual of ${\displaystyle M}$ is isometrically isomorphic to ${\displaystyle X^{\prime }/M^{\mathrm{\perp }}.}$ The dual of ${\displaystyle X/M}$ is isometrically isomorphic to ${\displaystyle M^{\mathrm{\perp }}.}$

The dual of a separable Banach space need not be separable, but:

Theorem—Let ${\displaystyle X}$ be a normed space. If ${\displaystyle X^{\prime }}$ is separable, then ${\displaystyle X}$ is separable.

When ${\displaystyle X^{\prime }}$ is separable, the above criterion for totality can be used for proving the existence of a countable total subset in ${\displaystyle X.}$

Weak topologies

The weak topology on a Banach space ${\displaystyle X}$ is the coarsest topology on ${\displaystyle X}$ for which all elements ${\displaystyle x^{\prime }}$ in the continuous dual space ${\displaystyle X^{\prime }}$ are continuous. The norm topology is therefore finer than the weak topology. It follows from the Hahn–Banach separation theorem that the weak topology is Hausdorff, and that a norm-closed convex subset of a Banach space is also weakly closed. A norm-continuous linear map between two Banach spaces ${\displaystyle X}$ and ${\displaystyle Y}$ is also weakly continuous, that is, continuous from the weak topology of ${\displaystyle X}$ to that of ${\displaystyle Y.}$

If ${\displaystyle X}$ is infinite-dimensional, there exist linear maps which are not continuous. The space ${\displaystyle X^{\ast }}$ of all linear maps from ${\displaystyle X}$ to the underlying field ${\displaystyle \mathbb{K}}$ (this space ${\displaystyle X^{\ast }}$ is called the algebraic dual space, to distinguish it from ${\displaystyle X^{\prime }}$ also induces a topology on ${\displaystyle X}$ which is finer than the weak topology, and much less used in functional analysis.

On a dual space ${\displaystyle X^{\prime },}$ there is a topology weaker than the weak topology of ${\displaystyle X^{\prime },}$ called the weak* topology. It is the coarsest topology on ${\displaystyle X^{\prime }}$ for which all evaluation maps ${\displaystyle x^{\prime }\in X^{\prime }\mapsto x^{\prime }(x),}$ where ${\displaystyle x}$ ranges over ${\displaystyle X,}$ are continuous. Its importance comes from the Banach–Alaoglu theorem.

Banach–Alaoglu theorem—Let ${\displaystyle X}$ be a normed vector space. Then the closed unit ball ${\displaystyle B=\{x\in X\mid \Vert x\Vert \le 1\}}$ of the dual space is compact in the weak* topology.

The Banach–Alaoglu theorem can be proved using Tychonoff's theorem about infinite products of compact Hausdorff spaces. When ${\displaystyle X}$ is separable, the unit ball ${\displaystyle B^{\prime }}$ of the dual is a metrizable compact in the weak* topology.

Examples of dual spaces

The dual of ${\displaystyle c_0}$ is isometrically isomorphic to ${\displaystyle {\ell }^1}$: for every bounded linear functional ${\displaystyle f}$ on ${\displaystyle c_0,}$ there is a unique element ${\displaystyle y=\{y_n\}\in {\ell }^1}$ such that $${\displaystyle f(x)=\sum _{n\in \mathbb{N}}{x}_n{y}_n,x=\{x_n\}\in c_0,\text{and}\Vert f{\Vert }_{(c_0{)}^{\prime }}=\Vert y{\Vert }_{{\ell }_1}.}$$

The dual of ${\displaystyle {\ell }^1}$ is isometrically isomorphic to ${\displaystyle {\ell }^{\mathrm{\infty }}}$. The dual of Lebesgue space ${\displaystyle L^p([0,1])}$ is isometrically isomorphic to ${\displaystyle L^q([0,1])}$ when ${\displaystyle 1\le p<\mathrm{\infty }}$ and ${\displaystyle \frac{1}{p}+\frac{1}{q}=1.}$

For every vector ${\displaystyle y}$ in a Hilbert space ${\displaystyle H,}$ the mapping $${\displaystyle x\in H\to f_y(x)=⟨x,y⟩}$$

defines a continuous linear functional ${\displaystyle f_y}$ on ${\displaystyle H.}$The Riesz representation theorem states that every continuous linear functional on ${\displaystyle H}$ is of the form ${\displaystyle f_y}$ for a uniquely defined vector ${\displaystyle y}$ in ${\displaystyle H.}$ The mapping ${\displaystyle y\in H\to f_y}$ is an antilinear isometric bijection from ${\displaystyle H}$ onto its dual ${\displaystyle H^{\prime }.}$ When the scalars are real, this map is an isometric isomorphism.

When ${\displaystyle K}$ is a compact Hausdorff topological space, the dual ${\displaystyle M(K)}$ of ${\displaystyle C(K)}$ is the space of Radon measures in the sense of Bourbaki. The subset ${\displaystyle P(K)}$ of ${\displaystyle M(K)}$ consisting of non-negative measures of mass 1 (probability measures) is a convex w*-closed subset of the unit ball of ${\displaystyle M(K).}$ The extreme points of ${\displaystyle P(K)}$ are the Dirac measures on ${\displaystyle K.}$ The set of Dirac measures on ${\displaystyle K,}$ equipped with the w*-topology, is homeomorphic to ${\displaystyle K.}$

Banach–Stone Theorem—If ${\displaystyle K}$ and ${\displaystyle L}$ are compact Hausdorff spaces and if ${\displaystyle C(K)}$ and ${\displaystyle C(L)}$ are isometrically isomorphic, then the topological spaces ${\displaystyle K}$ and ${\displaystyle L}$ are homeomorphic.

The result has been extended by Amir and Cambern to the case when the multiplicative Banach–Mazur distance between ${\displaystyle C(K)}$ and ${\displaystyle C(L)}$ is ${\displaystyle <2.}$ The theorem is no longer true when the distance is ${\displaystyle =2.}$

In the commutative Banach algebra ${\displaystyle C(K),}$ the maximal ideals are precisely kernels of Dirac measures on ${\displaystyle K,}$ $${\displaystyle I_x=\ker {\delta }_x=\{f\in C(K)\mid f(x)=0\},x\in K.}$$

More generally, by the Gelfand–Mazur theorem, the maximal ideals of a unital commutative Banach algebra can be identified with its characters—not merely as sets but as topological spaces: the former with the hull-kernel topology and the latter with the w*-topology. In this identification, the maximal ideal space can be viewed as a w*-compact subset of the unit ball in the dual ${\displaystyle A^{\prime }.}$

Theorem—If ${\displaystyle K}$ is a compact Hausdorff space, then the maximal ideal space ${\displaystyle \mathrm{\Xi }}$ of the Banach algebra ${\displaystyle C(K)}$ is homeomorphic to ${\displaystyle K.}$

Not every unital commutative Banach algebra is of the form ${\displaystyle C(K)}$ for some compact Hausdorff space ${\displaystyle K.}$ However, this statement holds if one places ${\displaystyle C(K)}$ in the smaller category of commutative C*-algebras. Gelfand's representation theorem for commutative C*-algebras states that every commutative unital C*-algebra ${\displaystyle A}$ is isometrically isomorphic to a ${\displaystyle C(K)}$ space. The Hausdorff compact space ${\displaystyle K}$ here is again the maximal ideal space, also called the spectrum of ${\displaystyle A}$ in the C*-algebra context.

Bidual

If ${\displaystyle X}$ is a normed space, the (continuous) dual ${\displaystyle X^{″}}$ of the dual ${\displaystyle X^{\prime }}$ is called the bidual or second dual of ${\displaystyle X.}$ For every normed space ${\displaystyle X,}$ there is a natural map, $${\displaystyle \{\begin{array}{l}{F}_X:X\to X^{″}\\ F_X(x)(f)=f(x)& \text{ for all }x\in X,\text{ and for all }f\in X^{\prime }\end{array}}$$

This defines ${\displaystyle F_X(x)}$ as a continuous linear functional on ${\displaystyle X^{\prime },}$ that is, an element of ${\displaystyle X^{″}.}$ The map ${\displaystyle F_X:x\to F_X(x)}$ is a linear map from ${\displaystyle X}$ to ${\displaystyle X^{″}.}$ As a consequence of the existence of a norming functional ${\displaystyle f}$ for every ${\displaystyle x\in X,}$ this map ${\displaystyle F_X}$ is isometric, thus injective.

For example, the dual of ${\displaystyle X=c_0}$ is identified with ${\displaystyle {\ell }^1,}$ and the dual of ${\displaystyle {\ell }^1}$ is identified with ${\displaystyle {\ell }^{\mathrm{\infty }},}$ the space of bounded scalar sequences. Under these identifications, ${\displaystyle F_X}$ is the inclusion map from ${\displaystyle c_0}$ to ${\displaystyle {\ell }^{\mathrm{\infty }}.}$ It is indeed isometric, but not onto.

If ${\displaystyle F_X}$ is surjective, then the normed space ${\displaystyle X}$ is called reflexive (see below). Being the dual of a normed space, the bidual ${\displaystyle X^{″}}$ is complete, therefore, every reflexive normed space is a Banach space.

Using the isometric embedding ${\displaystyle F_X,}$ it is customary to consider a normed space ${\displaystyle X}$ as a subset of its bidual. When ${\displaystyle X}$ is a Banach space, it is viewed as a closed linear subspace of ${\displaystyle X^{″}.}$ If ${\displaystyle X}$ is not reflexive, the unit ball of ${\displaystyle X}$ is a proper subset of the unit ball of ${\displaystyle X^{″}.}$ The Goldstine theorem states that the unit ball of a normed space is weakly*-dense in the unit ball of the bidual. In other words, for every ${\displaystyle x^{″}}$ in the bidual, there exists a net ${\displaystyle (x_i{)}_{i\in I}}$ in ${\displaystyle X}$ so that $${\displaystyle \underset{i\in I}{sup}\Vert x_i\Vert \le \Vert x^{″}\Vert ,x^{″}(f)=\underset{i}{lim}f(x_i),f\in X^{\prime }.}$$

The net may be replaced by a weakly*-convergent sequence when the dual ${\displaystyle X^{\prime }}$ is separable. On the other hand, elements of the bidual of ${\displaystyle {\ell }^1}$ that are not in ${\displaystyle {\ell }^1}$ cannot be weak*-limit of sequences in ${\displaystyle {\ell }^1,}$ since ${\displaystyle {\ell }^1}$ is weakly sequentially complete.

Banach's theorems

Here are the main general results about Banach spaces that go back to the time of Banach's book (Banach (1932)) and are related to the Baire category theorem. According to this theorem, a complete metric space (such as a Banach space, a Fréchet space or an F-space) cannot be equal to a union of countably many closed subsets with empty interiors. Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable Hamel basis is finite-dimensional.

Banach–Steinhaus Theorem—Let ${\displaystyle X}$ be a Banach space and ${\displaystyle Y}$ be a normed vector space. Suppose that ${\displaystyle F}$ is a collection of continuous linear operators from ${\displaystyle X}$ to ${\displaystyle Y.}$ The uniform boundedness principle states that if for all ${\displaystyle x}$ in ${\displaystyle X}$ we have ${\displaystyle \underset{T\in F}{sup}\Vert T(x){\Vert }_Y<\mathrm{\infty },}$ then ${\displaystyle \underset{T\in F}{sup}\Vert T{\Vert }_Y<\mathrm{\infty }.}$

The Banach–Steinhaus theorem is not limited to Banach spaces. It can be extended for example to the case where ${\displaystyle X}$ is a Fréchet space, provided the conclusion is modified as follows: under the same hypothesis, there exists a neighborhood ${\displaystyle U}$ of ${\displaystyle \mathbf{0}}$ in ${\displaystyle X}$ such that all ${\displaystyle T}$ in ${\displaystyle F}$ are uniformly bounded on ${\displaystyle U,}$ $${\displaystyle \underset{T\in F}{sup}\underset{x\in U}{sup}\Vert T(x){\Vert }_Y<\mathrm{\infty }.}$$

The Open Mapping Theorem—Let ${\displaystyle X}$ and ${\displaystyle Y}$ be Banach spaces and ${\displaystyle T:X\to Y}$ be a surjective continuous linear operator, then ${\displaystyle T}$ is an open map.

Corollary—Every one-to-one bounded linear operator from a Banach space onto a Banach space is an isomorphism.

The First Isomorphism Theorem for Banach spaces—Suppose that ${\displaystyle X}$ and ${\displaystyle Y}$ are Banach spaces and that ${\displaystyle T\in B(X,Y).}$ Suppose further that the range of ${\displaystyle T}$ is closed in ${\displaystyle Y.}$ Then ${\displaystyle X/\ker T}$ is isomorphic to ${\displaystyle T(X).}$

This result is a direct consequence of the preceding Banach isomorphism theorem and of the canonical factorization of bounded linear maps.

Corollary—If a Banach space ${\displaystyle X}$ is the internal direct sum of closed subspaces ${\displaystyle M_1,\dots ,M_n,}$ then ${\displaystyle X}$ is isomorphic to ${\displaystyle M_1\oplus \cdots \oplus M_n.}$

This is another consequence of Banach's isomorphism theorem, applied to the continuous bijection from ${\displaystyle M_1\oplus \cdots \oplus M_n}$ onto ${\displaystyle X}$ sending ${\displaystyle m_1,\cdots ,m_n}$ to the sum ${\displaystyle m_1+\cdots +m_n.}$

The Closed Graph Theorem—Let ${\displaystyle T:X\to Y}$ be a linear mapping between Banach spaces. The graph of ${\displaystyle T}$ is closed in ${\displaystyle X\times Y}$ if and only if ${\displaystyle T}$ is continuous.

Reflexivity

The normed space ${\displaystyle X}$ is called reflexive when the natural map $${\displaystyle \{\begin{array}{l}{F}_X:X\to X^{″}\\ F_X(x)(f)=f(x)& \text{ for all }x\in X,\text{ and for all }f\in X^{\prime }\end{array}}$$ is surjective. Reflexive normed spaces are Banach spaces.

Theorem—If ${\displaystyle X}$ is a reflexive Banach space, every closed subspace of ${\displaystyle X}$ and every quotient space of ${\displaystyle X}$ are reflexive.

This is a consequence of the Hahn–Banach theorem. Further, by the open mapping theorem, if there is a bounded linear operator from the Banach space ${\displaystyle X}$ onto the Banach space ${\displaystyle Y,}$ then ${\displaystyle Y}$ is reflexive.

Theorem—If ${\displaystyle X}$ is a Banach space, then ${\displaystyle X}$ is reflexive if and only if ${\displaystyle X^{\prime }}$ is reflexive.

Corollary—Let ${\displaystyle X}$ be a reflexive Banach space. Then ${\displaystyle X}$ is separable if and only if ${\displaystyle X^{\prime }}$ is separable.

Indeed, if the dual ${\displaystyle Y^{\prime }}$ of a Banach space ${\displaystyle Y}$ is separable, then ${\displaystyle Y}$ is separable. If ${\displaystyle X}$ is reflexive and separable, then the dual of ${\displaystyle X^{\prime }}$ is separable, so ${\displaystyle X^{\prime }}$ is separable.

Theorem—Suppose that ${\displaystyle X_1,\dots ,X_n}$ are normed spaces and that ${\displaystyle X=X_1\oplus \cdots \oplus X_n.}$ Then ${\displaystyle X}$ is reflexive if and only if each ${\displaystyle X_j}$ is reflexive.

Hilbert spaces are reflexive. The ${\displaystyle L^p}$ spaces are reflexive when ${\displaystyle 1<p<\mathrm{\infty }.}$ More generally, uniformly convex spaces are reflexive, by the Milman–Pettis theorem. The spaces ${\displaystyle c_0,{\ell }^1,L^1([0,1]),C([0,1])}$ are not reflexive. In these examples of non-reflexive spaces ${\displaystyle X,}$ the bidual ${\displaystyle X^{″}}$ is "much larger" than ${\displaystyle X.}$ Namely, under the natural isometric embedding of ${\displaystyle X}$ into ${\displaystyle X^{″}}$ given by the Hahn–Banach theorem, the quotient ${\displaystyle X^{″}/X}$ is infinite-dimensional, and even nonseparable. However, Robert C. James has constructed an example of a non-reflexive space, usually called "the James space" and denoted by ${\displaystyle J,}$ such that the quotient ${\displaystyle J^{″}/J}$ is one-dimensional. Furthermore, this space ${\displaystyle J}$ is isometrically isomorphic to its bidual.

Theorem—A Banach space ${\displaystyle X}$ is reflexive if and only if its unit ball is compact in the weak topology.

When ${\displaystyle X}$ is reflexive, it follows that all closed and bounded convex subsets of ${\displaystyle X}$ are weakly compact. In a Hilbert space ${\displaystyle H,}$ the weak compactness of the unit ball is very often used in the following way: every bounded sequence in ${\displaystyle H}$ has weakly convergent subsequences.

Weak compactness of the unit ball provides a tool for finding solutions in reflexive spaces to certain optimization problems. For example, every convex continuous function on the unit ball ${\displaystyle B}$ of a reflexive space attains its minimum at some point in ${\displaystyle B.}$

As a special case of the preceding result, when ${\displaystyle X}$ is a reflexive space over ${\displaystyle \mathbb{R},}$ every continuous linear functional ${\displaystyle f}$ in ${\displaystyle X^{\prime }}$ attains its maximum ${\displaystyle \Vert f\Vert }$ on the unit ball of ${\displaystyle X.}$ The following theorem of Robert C. James provides a converse statement.

James' Theorem—For a Banach space the following two properties are equivalent:

• ${\displaystyle X}$ is reflexive.
• for all ${\displaystyle f}$ in ${\displaystyle X^{\prime }}$ there exists ${\displaystyle x\in X}$ with ${\displaystyle \Vert x\Vert \le 1,}$ so that ${\displaystyle f(x)=\Vert f\Vert .}$

The theorem can be extended to give a characterization of weakly compact convex sets.

On every non-reflexive Banach space ${\displaystyle X,}$ there exist continuous linear functionals that are not norm-attaining. However, the Bishop–Phelps theorem states that norm-attaining functionals are norm dense in the dual ${\displaystyle X^{\prime }}$ of ${\displaystyle X.}$

Weak convergences of sequences

A sequence ${\displaystyle \{x_n\}}$ in a Banach space ${\displaystyle X}$ is weakly convergent to a vector ${\displaystyle x\in X}$ if ${\displaystyle \{f(x_n)\}}$ converges to ${\displaystyle f(x)}$ for every continuous linear functional ${\displaystyle f}$ in the dual ${\displaystyle X^{\prime }.}$ The sequence ${\displaystyle \{x_n\}}$ is a weakly Cauchy sequence if ${\displaystyle \{f(x_n)\}}$ converges to a scalar limit ${\displaystyle L(f)}$ for every ${\displaystyle f}$ in ${\displaystyle X^{\prime }.}$ A sequence ${\displaystyle \{f_n\}}$ in the dual ${\displaystyle X^{\prime }}$ is weakly* convergent to a functional ${\displaystyle f\in X^{\prime }}$ if ${\displaystyle f_n(x)}$ converges to ${\displaystyle f(x)}$ for every ${\displaystyle x}$ in ${\displaystyle X.}$ Weakly Cauchy sequences, weakly convergent and weakly* convergent sequences are norm bounded, as a consequence of the Banach–Steinhaus theorem.

When the sequence ${\displaystyle \{x_n\}}$ in ${\displaystyle X}$ is a weakly Cauchy sequence, the limit ${\displaystyle L}$ above defines a bounded linear functional on the dual ${\displaystyle X^{\prime },}$ that is, an element ${\displaystyle L}$ of the bidual of ${\displaystyle X,}$ and ${\displaystyle L}$ is the limit of ${\displaystyle \{x_n\}}$ in the weak*-topology of the bidual. The Banach space ${\displaystyle X}$ is weakly sequentially complete if every weakly Cauchy sequence is weakly convergent in ${\displaystyle X.}$ It follows from the preceding discussion that reflexive spaces are weakly sequentially complete.

Theorem —For every measure ${\displaystyle \mu ,}$ the space ${\displaystyle L^1(\mu )}$ is weakly sequentially complete.

An orthonormal sequence in a Hilbert space is a simple example of a weakly convergent sequence, with limit equal to the ${\displaystyle \mathbf{0}}$ vector. The unit vector basis of ${\displaystyle {\ell }^p}$ for ${\displaystyle 1<p<\mathrm{\infty },}$ or of ${\displaystyle c_0,}$ is another example of a weakly null sequence, that is, a sequence that converges weakly to ${\displaystyle \mathbf{0}.}$ For every weakly null sequence in a Banach space, there exists a sequence of convex combinations of vectors from the given sequence that is norm-converging to ${\displaystyle \mathbf{0}.}$

The unit vector basis of ${\displaystyle {\ell }^1}$ is not weakly Cauchy. Weakly Cauchy sequences in ${\displaystyle {\ell }^1}$ are weakly convergent, since ${\displaystyle L^1}$-spaces are weakly sequentially complete. Actually, weakly convergent sequences in ${\displaystyle {\ell }^1}$ are norm convergent. This means that ${\displaystyle {\ell }^1}$ satisfies Schur's property.

Results involving the 𝓁1 basis

Weakly Cauchy sequences and the ${\displaystyle {\ell }^1}$ basis are the opposite cases of the dichotomy established in the following deep result of Haskell P. Rosenthal.

Theorem—Let ${\displaystyle \{x_n{\}}_{n\in \mathbb{N}}}$ be a bounded sequence in a Banach space. Either ${\displaystyle \{x_n{\}}_{n\in \mathbb{N}}}$ has a weakly Cauchy subsequence, or it admits a subsequence equivalent to the standard unit vector basis of ${\displaystyle {\ell }^1.}$

A complement to this result is due to Odell and Rosenthal (1975).

Theorem—Let ${\displaystyle X}$ be a separable Banach space. The following are equivalent:

• The space ${\displaystyle X}$ does not contain a closed subspace isomorphic to ${\displaystyle {\ell }^1.}$
• Every element of the bidual ${\displaystyle X^{″}}$ is the weak*-limit of a sequence ${\displaystyle \{x_n\}}$ in ${\displaystyle X.}$

By the Goldstine theorem, every element of the unit ball ${\displaystyle B^{″}}$ of ${\displaystyle X^{″}}$ is weak*-limit of a net in the unit ball of ${\displaystyle X.}$ When ${\displaystyle X}$ does not contain ${\displaystyle {\ell }^1,}$ every element of ${\displaystyle B^{″}}$ is weak*-limit of a sequence in the unit ball of ${\displaystyle X.}$

When the Banach space ${\displaystyle X}$ is separable, the unit ball of the dual ${\displaystyle X^{\prime },}$ equipped with the weak*-topology, is a metrizable compact space ${\displaystyle K,}$ and every element ${\displaystyle x^{″}}$ in the bidual ${\displaystyle X^{″}}$ defines a bounded function on ${\displaystyle K}$: $${\displaystyle x^{\prime }\in K\mapsto x^{″}(x^{\prime }),|x^{″}(x^{\prime })|\le \Vert x^{″}\Vert .}$$

This function is continuous for the compact topology of ${\displaystyle K}$ if and only if ${\displaystyle x^{″}}$ is actually in ${\displaystyle X,}$ considered as subset of ${\displaystyle X^{″}.}$ Assume in addition for the rest of the paragraph that ${\displaystyle X}$ does not contain ${\displaystyle {\ell }^1.}$ By the preceding result of Odell and Rosenthal, the function ${\displaystyle x^{″}}$ is the pointwise limit on ${\displaystyle K}$ of a sequence ${\displaystyle \{x_n\}\subseteq X}$ of continuous functions on ${\displaystyle K,}$ it is therefore a first Baire class function on ${\displaystyle K.}$ The unit ball of the bidual is a pointwise compact subset of the first Baire class on ${\displaystyle K.}$

Sequences, weak and weak* compactness

When ${\displaystyle X}$ is separable, the unit ball of the dual is weak*-compact by the Banach–Alaoglu theorem and metrizable for the weak* topology, hence every bounded sequence in the dual has weakly* convergent subsequences. This applies to separable reflexive spaces, but more is true in this case, as stated below.

The weak topology of a Banach space ${\displaystyle X}$ is metrizable if and only if ${\displaystyle X}$ is finite-dimensional. If the dual ${\displaystyle X^{\prime }}$ is separable, the weak topology of the unit ball of ${\displaystyle X}$ is metrizable. This applies in particular to separable reflexive Banach spaces. Although the weak topology of the unit ball is not metrizable in general, one can characterize weak compactness using sequences.

Eberlein–Šmulian theorem—A set ${\displaystyle A}$ in a Banach space is relatively weakly compact if and only if every sequence ${\displaystyle \{a_n\}}$ in ${\displaystyle A}$ has a weakly convergent subsequence.

A Banach space ${\displaystyle X}$ is reflexive if and only if each bounded sequence in ${\displaystyle X}$ has a weakly convergent subsequence.

A weakly compact subset ${\displaystyle A}$ in ${\displaystyle {\ell }^1}$ is norm-compact. Indeed, every sequence in ${\displaystyle A}$ has weakly convergent subsequences by Eberlein–Šmulian, that are norm convergent by the Schur property of ${\displaystyle {\ell }^1.}$

Type and cotype

A way to classify Banach spaces is through the probabilistic notion of type and cotype, these two measure how far a Banach space is from a Hilbert space.

Schauder bases

A Schauder basis in a Banach space ${\displaystyle X}$ is a sequence ${\displaystyle \{e_n{\}}_{n\ge 0}}$ of vectors in ${\displaystyle X}$ with the property that for every vector ${\displaystyle x\in X,}$ there exist uniquely defined scalars ${\displaystyle \{x_n{\}}_{n\ge 0}}$ depending on ${\displaystyle x,}$ such that $${\displaystyle x=\sum _{n=0}^{\mathrm{\infty }}{x}_n{e}_n,\mathit{\text{i.e.,}}x=\underset{n}{lim}{P}_n(x),P_n(x):=\sum _{k=0}^n{x}_k{e}_k.}$$

Banach spaces with a Schauder basis are necessarily separable, because the countable set of finite linear combinations with rational coefficients (say) is dense.

It follows from the Banach–Steinhaus theorem that the linear mappings ${\displaystyle \{P_n\}}$ are uniformly bounded by some constant ${\displaystyle C.}$ Let ${\displaystyle \{e_n^{\ast }\}}$ denote the coordinate functionals which assign to every ${\displaystyle x}$ in ${\displaystyle X}$ the coordinate ${\displaystyle x_n}$ of ${\displaystyle x}$ in the above expansion. They are called biorthogonal functionals. When the basis vectors have norm ${\displaystyle 1,}$ the coordinate functionals ${\displaystyle \{e_n^{\ast }\}}$ have norm ${\displaystyle \le 2C}$ in the dual of ${\displaystyle X.}$

Most classical separable spaces have explicit bases. The Haar system ${\displaystyle \{h_n\}}$ is a basis for ${\displaystyle L^p([0,1])}$ when ${\displaystyle 1\le p<\mathrm{\infty }.}$ The trigonometric system is a basis in ${\displaystyle L^p(\mathbf{T})}$ when ${\displaystyle 1<p<\mathrm{\infty }.}$ The Schauder system is a basis in the space ${\displaystyle C([0,1]).}$ The question of whether the disk algebra ${\displaystyle A(\mathbf{D})}$ has a basis remained open for more than forty years, until Bočkarev showed in 1974 that ${\displaystyle A(\mathbf{D})}$ admits a basis constructed from the Franklin system.

Since every vector ${\displaystyle x}$ in a Banach space ${\displaystyle X}$ with a basis is the limit of ${\displaystyle P_n(x),}$ with ${\displaystyle P_n}$ of finite rank and uniformly bounded, the space ${\displaystyle X}$ satisfies the bounded approximation property. The first example by Enflo of a space failing the approximation property was at the same time the first example of a separable Banach space without a Schauder basis.

Robert C. James characterized reflexivity in Banach spaces with a basis: the space ${\displaystyle X}$ with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete. In this case, the biorthogonal functionals form a basis of the dual of ${\displaystyle X.}$

Tensor product

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be two ${\displaystyle \mathbb{K}}$-vector spaces. The tensor product ${\displaystyle X\otimes Y}$ of ${\displaystyle X}$ and ${\displaystyle Y}$ is a ${\displaystyle \mathbb{K}}$-vector space ${\displaystyle Z}$ with a bilinear mapping ${\displaystyle T:X\times Y\to Z}$ which has the following universal property:

If ${\displaystyle T_1:X\times Y\to Z_1}$ is any bilinear mapping into a ${\displaystyle \mathbb{K}}$-vector space ${\displaystyle Z_1,}$ then there exists a unique linear mapping ${\displaystyle f:Z\to Z_1}$ such that ${\displaystyle T_1=f\circ T.}$

The image under ${\displaystyle T}$ of a couple ${\displaystyle (x,y)}$ in ${\displaystyle X\times Y}$ is denoted by ${\displaystyle x\otimes y,}$ and called a simple tensor. Every element ${\displaystyle z}$ in ${\displaystyle X\otimes Y}$ is a finite sum of such simple tensors.

There are various norms that can be placed on the tensor product of the underlying vector spaces, amongst others the projective cross norm and injective cross norm introduced by A. Grothendieck in 1955.

In general, the tensor product of complete spaces is not complete again. When working with Banach spaces, it is customary to say that the projective tensor product of two Banach spaces ${\displaystyle X}$ and ${\displaystyle Y}$ is the completion ${\displaystyle X{\hat{\otimes }}_{\pi }Y}$ of the algebraic tensor product ${\displaystyle X\otimes Y}$ equipped with the projective tensor norm, and similarly for the injective tensor product ${\displaystyle X{\hat{\otimes }}_{\epsilon }Y.}$ Grothendieck proved in particular that

$${\displaystyle \begin{array}{cc}C(K){\hat{\otimes }}_{\epsilon }Y& \simeq C(K,Y),\\ L^1([0,1]){\hat{\otimes }}_{\pi }Y& \simeq L^1([0,1],Y),\end{array}}$$ where ${\displaystyle K}$ is a compact Hausdorff space, ${\displaystyle C(K,Y)}$ the Banach space of continuous functions from ${\displaystyle K}$ to ${\displaystyle Y}$ and ${\displaystyle L^1([0,1],Y)}$ the space of Bochner-measurable and integrable functions from ${\displaystyle [0,1]}$ to ${\displaystyle Y,}$ and where the isomorphisms are isometric. The two isomorphisms above are the respective extensions of the map sending the tensor ${\displaystyle f\otimes y}$ to the vector-valued function ${\displaystyle s\in K\to f(s)y\in Y.}$

Tensor products and the approximation property

Let ${\displaystyle X}$ be a Banach space. The tensor product ${\displaystyle X^{\prime }{\hat{\otimes }}_{\epsilon }X}$ is identified isometrically with the closure in ${\displaystyle B(X)}$ of the set of finite rank operators. When ${\displaystyle X}$ has the approximation property, this closure coincides with the space of compact operators on ${\displaystyle X.}$

For every Banach space ${\displaystyle Y,}$ there is a natural norm ${\displaystyle 1}$ linear map $${\displaystyle Y{\hat{\otimes }}_{\pi }X\to Y{\hat{\otimes }}_{\epsilon }X}$$ obtained by extending the identity map of the algebraic tensor product. Grothendieck related the approximation problem to the question of whether this map is one-to-one when ${\displaystyle Y}$ is the dual of ${\displaystyle X.}$ Precisely, for every Banach space ${\displaystyle X,}$ the map $${\displaystyle X^{\prime }{\hat{\otimes }}_{\pi }X⟶X^{\prime }{\hat{\otimes }}_{\epsilon }X}$$ is one-to-one if and only if ${\displaystyle X}$ has the approximation property.

Grothendieck conjectured that ${\displaystyle X{\hat{\otimes }}_{\pi }Y}$ and ${\displaystyle X{\hat{\otimes }}_{\epsilon }Y}$ must be different whenever ${\displaystyle X}$ and ${\displaystyle Y}$ are infinite-dimensional Banach spaces. This was disproved by Gilles Pisier in 1983. Pisier constructed an infinite-dimensional Banach space ${\displaystyle X}$ such that ${\displaystyle X{\hat{\otimes }}_{\pi }X}$ and ${\displaystyle X{\hat{\otimes }}_{\epsilon }X}$ are equal. Furthermore, just as Enflo's example, this space ${\displaystyle X}$ is a "hand-made" space that fails to have the approximation property. On the other hand, Szankowski proved that the classical space ${\displaystyle B({\ell }^2)}$ does not have the approximation property.

Some classification results

Characterizations of Hilbert space among Banach spaces

A necessary and sufficient condition for the norm of a Banach space ${\displaystyle X}$ to be associated to an inner product is the parallelogram identity:

Parallelogram identity—for all ${\displaystyle x,y\in X:\Vert x+y{\Vert }^2+\Vert x-y{\Vert }^2=2(\Vert x{\Vert }^2+\Vert y{\Vert }^2).}$

It follows, for example, that the Lebesgue space ${\displaystyle L^p([0,1])}$ is a Hilbert space only when ${\displaystyle p=2.}$ If this identity is satisfied, the associated inner product is given by the polarization identity. In the case of real scalars, this gives: $${\displaystyle ⟨x,y⟩=\frac{1}{4}(\Vert x+y{\Vert }^2-\Vert x-y{\Vert }^2).}$$