Linear operators, isomorphisms If X and Y are normed spaces over the same
ground field \mathbb{K}, the set of all
continuous \mathbb{K}-linear maps T : X \to Y is denoted by B(X, Y). In infinite-dimensional spaces, not all linear maps are continuous. A linear mapping from a normed space X to another normed space is continuous if and only if it is
bounded on the closed
unit ball of X. Thus, the vector space B(X, Y) can be given the
operator norm \|T\| = \sup \{\|Tx\|_Y \mid x\in X,\ \|x\|_X \leq 1\}. For Y a Banach space, the space 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 B(X,Y) reappears as a natural
bifunctor. If X is a Banach space, the space B(X) = B(X, X) forms a unital
Banach algebra; the multiplication operation is given by the composition of linear maps. If X and Y are normed spaces, they are
isomorphic normed spaces if there exists a linear bijection T : X \to Y such that T and its inverse T^{-1} are continuous. If one of the two spaces X or Y is complete (or
reflexive,
separable, etc.) then so is the other space. Two normed spaces X and Y are
isometrically isomorphic if in addition, T is an
isometry, that is, \|T(x)\| = \|x\| for every x in X. The
Banach–Mazur distance d(X, Y) between two isomorphic but not isometric spaces X and Y gives a measure of how much the two spaces X and 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 \R or \Complex) 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 f : X \to \R is a
subadditive function (such as a norm, a
sublinear function, or real linear functional), then f is
continuous at the origin if and only if f is
uniformly continuous on all of X; and if in addition f(0) = 0 then f is continuous if and only if its
absolute value |f| : X \to [0, \infty) is continuous, which happens if and only if \{x \in X \mid |f(x)| is an open subset of X. And very importantly for applying the
Hahn–Banach theorem, a linear functional f is continuous if and only if this is true of its
real part \operatorname{Re} f and moreover, \|\operatorname{Re} f\| = \|f\| and
the real part \operatorname{Re} f completely determines f, which is why the Hahn–Banach theorem is often stated only for real linear functionals. Also, a linear functional f on X is continuous if and only if the
seminorm |f| is continuous, which happens if and only if there exists a continuous seminorm p : X \to \R such that |f| \leq p; this last statement involving the linear functional f and seminorm p is encountered in many versions of the Hahn–Banach theorem.
Basic notions The Cartesian product X \times Y of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are commonly used, such as \|(x, y)\|_1 = \|x\| + \|y\|, \qquad \|(x, y)\|_\infty = \max(\|x\|, \|y\|) which correspond (respectively) to the
coproduct and
product in the category of Banach spaces and short maps (discussed above). The quotient map from X onto X / M, sending x \in X to its class x + M, is linear, onto, and of norm 1, except when M = X, in which case the quotient is the null space. The closed linear subspace M of X is said to be a
complemented subspace of X if M is the
range of a
surjective bounded linear
projection P : X \to M. In this case, the space X is isomorphic to the direct sum of M and \ker P, the kernel of the projection P. Suppose that X and Y are Banach spaces and that T \in B(X, Y). There exists a canonical factorization of T as of Banach spaces include: the
Lp spaces L^p and their special cases, the
sequence spaces \ell^p that consist of scalar sequences indexed by
natural numbers \N; among them, the space \ell^1 of
absolutely summable sequences and the space \ell^2 of square summable sequences; the space c_0 of sequences tending to zero and the space \ell^{\infty} of bounded sequences; the space C(K) of continuous scalar functions on a compact Hausdorff space K, equipped with the max norm, \|f\|_{C(K)} = \max \{ |f(x)| \mid x \in K \}, \quad f \in C(K). According to the
Banach–Mazur theorem, every Banach space is isometrically isomorphic to a subspace of some C(K). For every separable Banach space X, there is a closed subspace M of \ell^1 such that X := \ell^1 / M. Any
Hilbert space serves as an example of a Banach space. A Hilbert space H on \mathbb{K} = \Reals, \Complex is complete for a norm of the form \|x\|_H = \sqrt{\langle x, x \rangle}, where \langle \cdot, \cdot \rangle : H \times H \to \mathbb{K} is the
inner product, linear in its first argument that satisfies the following: \begin{align} \langle y, x \rangle &= \overline{\langle x, y \rangle}, \quad \text{ for all } x, y \in H \\ \langle x, x \rangle & \geq 0, \quad \text{ for all } x \in H \\ \langle x,x \rangle = 0 \text{ if and only if } x &= 0. \end{align} For example, the space L^2 is a Hilbert space. The
Hardy spaces, the
Sobolev spaces are examples of Banach spaces that are related to 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 A over \mathbb{K} = \R or \Complex, together with a structure of
algebra over \mathbb{K}, such that the product map A \times A \ni (a, b) \mapsto ab \in A is continuous. An equivalent norm on A can be found so that \|ab\| \leq \|a\| \|b\| for all a, b \in A.
Examples • The Banach space C(K) with the pointwise product, is a Banach algebra. • The
disk algebra A(\mathbf{D}) consists of functions
holomorphic in the open unit disk D \subseteq \Complex and continuous on its
closure: \overline{\mathbf{D}}. Equipped with the max norm on \overline{\mathbf{D}}, the disk algebra A(\mathbf{D}) is a closed subalgebra of C\left(\overline{\mathbf{D}}\right). • The
Wiener algebra A(\mathbf{T}) is the algebra of functions on the unit circle \mathbf{T} with absolutely convergent Fourier series. Via the map associating a function on \mathbf{T} to the sequence of its Fourier coefficients, this algebra is isomorphic to the Banach algebra \ell^1(Z), where the product is the
convolution of sequences. • For every Banach space X, the space B(X) of bounded linear operators on X, with the composition of maps as product, is a Banach algebra. • A
C*-algebra is a complex Banach algebra A with an
antilinear involution a \mapsto a^* such that \|a^* a\| = \|a\|^2. The space B(H) of bounded linear operators on a Hilbert space 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 B(H). The space C(K) of complex continuous functions on a compact Hausdorff space K is an example of commutative C*-algebra, where the involution associates to every function f its
complex conjugate \overline{f}.
Dual space If X is a normed space and \mathbb{K} the underlying
field (either the
reals or the
complex numbers), the
continuous dual space is the space of continuous linear maps from X into \mathbb{K}, or
continuous linear functionals. The notation for the continuous dual is X' = B(X, \mathbb{K}) in this article. Since \mathbb{K} is a Banach space (using the
absolute value as norm), the dual X' is a Banach space, for every normed space 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. {{math theorem|name=Hahn–Banach theorem|math_statement=Let X be a
vector space over the field \mathbb{K} = \R, \Complex. Let further • Y \subseteq X be a
linear subspace, • p : X \to \R be a
sublinear function and • f : Y \to \mathbb{K} be a
linear functional so that \operatorname{Re}(f(y)) \leq p(y) for all y \in Y. Then, there exists a linear functional F : X \to \mathbb{K} so that F\big\vert_Y = f, \quad \text{ and } \quad \text{ for all } x \in X, \ \ \operatorname{Re}(F(x)) \leq 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 x in a normed space X, there exists a continuous linear functional f on X such that f(x) = \|x\|_X, \quad \|f\|_{X'} \leq 1. When x is not equal to the \mathbf{0} vector, the functional f must have norm one, and is called a
norming functional for 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 S in a Banach space X is
total if the
linear span of S is
dense in X. The subset S is total in X if and only if the only continuous linear functional that vanishes on S is the \mathbf{0} functional: this equivalence follows from the Hahn–Banach theorem. If X is the direct sum of two closed linear subspaces M and N, then the dual X' of X is isomorphic to the direct sum of the duals of M and N. If M is a closed linear subspace in X, one can associate the M in the dual, M^{\bot} = \{ x' \in X \mid x'(m) = 0 \text{ for all } m \in M \}. The orthogonal M^{\bot} is a closed linear subspace of the dual. The dual of M is isometrically isomorphic to X' / M^{\bot}. The dual of X / M is isometrically isomorphic to M^{\bot}. The dual of a separable Banach space need not be separable, but: When X' is separable, the above criterion for totality can be used for proving the existence of a countable total subset in X.
Weak topologies The
weak topology on a Banach space X is the
coarsest topology on X for which all elements x' in the continuous dual space X' 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 X and Y is also
weakly continuous, that is, continuous from the weak topology of X to that of Y. If X is infinite-dimensional, there exist linear maps which are not continuous. The space X^* of all linear maps from X to the underlying field \mathbb{K} (this space X^* is called the
algebraic dual space, to distinguish it from X' also induces a topology on X which is
finer than the weak topology, and much less used in functional analysis. On a dual space X', there is a topology weaker than the weak topology of X', called the
weak* topology. It is the coarsest topology on X' for which all evaluation maps x' \in X' \mapsto x'(x), where x ranges over X, are continuous. Its importance comes from the
Banach–Alaoglu theorem. {{math theorem|name=
Banach–Alaoglu theorem|math_statement=Let X be a
normed vector space. Then the
closed unit ball B = \{x \in X \mid \|x\| \leq 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 X is separable, the unit ball B' of the dual is a
metrizable compact in the weak* topology.
Examples of dual spaces The dual of c_0 is isometrically isomorphic to \ell^1: for every bounded linear functional f on c_0, there is a unique element y = \{y_n\} \in \ell^1 such that f(x) = \sum_{n \in \N} x_n y_n, \qquad x = \{x_n\} \in c_0, \ \ \text{and} \ \ \|f\|_{(c_0)'} = \|y\|_{\ell_1}. The dual of \ell^1 is isometrically isomorphic to \ell^{\infty}. The dual of
Lebesgue space L^p([0, 1]) is isometrically isomorphic to L^q([0, 1]) when 1 \leq p and \frac{1}{p} + \frac{1}{q} = 1. For every vector y in a Hilbert space H, the mapping x \in H \to f_y(x) = \langle x, y \rangle defines a continuous linear functional f_y on H.The
Riesz representation theorem states that every continuous linear functional on H is of the form f_y for a uniquely defined vector y in H. The mapping y \in H \to f_y is an
antilinear isometric bijection from H onto its dual H'. When the scalars are real, this map is an isometric isomorphism. When K is a compact Hausdorff topological space, the dual M(K) of C(K) is the space of
Radon measures in the sense of Bourbaki. The subset P(K) of M(K) consisting of non-negative measures of mass 1 (
probability measures) is a convex w*-closed subset of the unit ball of M(K). The
extreme points of P(K) are the
Dirac measures on K. The set of Dirac measures on K, equipped with the w*-topology, is
homeomorphic to K. The result has been extended by Amir and Cambern to the case when the multiplicative
Banach–Mazur distance between C(K) and C(L) is The theorem is no longer true when the distance is = 2. In the commutative
Banach algebra C(K), the
maximal ideals are precisely kernels of Dirac measures on K, I_x = \ker \delta_x = \{f \in C(K) \mid f(x) = 0\}, \quad 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 A'. Not every unital commutative Banach algebra is of the form C(K) for some compact Hausdorff space K. However, this statement holds if one places 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 A is isometrically isomorphic to a C(K) space. The Hausdorff compact space K here is again the maximal ideal space, also called the
spectrum of A in the C*-algebra context.
Bidual If X is a normed space, the (continuous) dual X'' of the dual X' is called the '
or ' of X. For every normed space X, there is a natural map, \begin{cases} F_X\colon X \to X'' \\ F_X(x) (f) = f(x) & \text{ for all } x \in X, \text{ and for all } f \in X' \end{cases} This defines F_X(x) as a continuous linear functional on X', that is, an element of X
. The map F_X \colon x \to F_X(x) is a linear map from X to X. As a consequence of the existence of a
norming functional f for every x \in X, this map F_X is isometric, thus
injective. For example, the dual of X = c_0 is identified with \ell^1, and the dual of \ell^1 is identified with \ell^{\infty}, the space of bounded scalar sequences. Under these identifications, F_X is the inclusion map from c_0 to \ell^{\infty}. It is indeed isometric, but not onto. If F_X is
surjective, then the normed space X is called
reflexive (see
below). Being the dual of a normed space, the bidual X'' is complete, therefore, every reflexive normed space is a Banach space. Using the isometric embedding F_X, it is customary to consider a normed space X as a subset of its bidual. When X is a Banach space, it is viewed as a closed linear subspace of X
. If X is not reflexive, the unit ball of X is a proper subset of the unit ball of 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 x'' in the bidual, there exists a
net (x_i)_{i \in I} in X so that \sup_{i \in I} \|x_i\| \leq \|x
\|, \ \ x(f) = \lim_i f(x_i), \quad f \in X'. The net may be replaced by a weakly*-convergent sequence when the dual X' is separable. On the other hand, elements of the bidual of \ell^1 that are not in \ell^1 cannot be weak*-limit of in \ell^1, since \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 () 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. {{math theorem|name=
Banach–Steinhaus Theorem|math_statement=Let X be a Banach space and Y be a
normed vector space. Suppose that F is a collection of continuous linear operators from X to Y. The uniform boundedness principle states that if for all x in X we have \sup_{T \in F} \|T(x)\|_Y then \sup_{T \in F} \|T\|_Y }} The Banach–Steinhaus theorem is not limited to Banach spaces. It can be extended for example to the case where X is a
Fréchet space, provided the conclusion is modified as follows: under the same hypothesis, there exists a neighborhood U of \mathbf{0} in X such that all T in F are uniformly bounded on U, \sup_{T \in F} \sup_{x \in U} \; \|T(x)\|_Y This result is a direct consequence of the preceding
Banach isomorphism theorem and of the canonical factorization of bounded linear maps. This is another consequence of Banach's isomorphism theorem, applied to the continuous bijection from M_1 \oplus \cdots \oplus M_n onto X sending m_1, \cdots, m_n to the sum m_1 + \cdots + m_n.
Reflexivity The normed space X is called
reflexive when the natural map \begin{cases} F_X : X \to X'' \\ F_X(x) (f) = f(x) & \text{ for all } x \in X, \text{ and for all } f \in X'\end{cases} is surjective. Reflexive normed spaces are Banach spaces. 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 X onto the Banach space Y, then Y is reflexive. Indeed, if the dual Y' of a Banach space Y is separable, then Y is separable. If X is reflexive and separable, then the dual of X' is separable, so X' is separable. Hilbert spaces are reflexive. The L^p spaces are reflexive when 1 More generally,
uniformly convex spaces are reflexive, by the
Milman–Pettis theorem. The spaces c_0, \ell^1, L^1([0, 1]), C([0, 1]) are not reflexive. In these examples of non-reflexive spaces X, the bidual X'' is "much larger" than X. Namely, under the natural isometric embedding of X into X
given by the Hahn–Banach theorem, the quotient 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 J, such that the quotient J'' / J is one-dimensional. Furthermore, this space J is isometrically isomorphic to its bidual. When X is reflexive, it follows that all closed and bounded
convex subsets of X are weakly compact. In a Hilbert space H, the weak compactness of the unit ball is very often used in the following way: every bounded sequence in 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 B of a reflexive space attains its minimum at some point in B. As a special case of the preceding result, when X is a reflexive space over \R, every continuous linear functional f in X' attains its maximum \|f\| on the unit ball of X. The following
theorem of Robert C. James provides a converse statement. The theorem can be extended to give a characterization of weakly compact convex sets. On every non-reflexive Banach space 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 X' of X.
Weak convergences of sequences A sequence \{x_n\} in a Banach space X is
weakly convergent to a vector x \in X if \{f(x_n)\} converges to f(x) for every continuous linear functional f in the dual X'. The sequence \{x_n\} is a
weakly Cauchy sequence if \{f(x_n)\} converges to a scalar limit L(f) for every f in X'. A sequence \{f_n\} in the dual X' is
weakly* convergent to a functional f \in X' if f_n(x) converges to f(x) for every x in X. Weakly Cauchy sequences, weakly convergent and weakly* convergent sequences are norm bounded, as a consequence of the
Banach–Steinhaus theorem. When the sequence \{x_n\} in X is a weakly Cauchy sequence, the limit L above defines a bounded linear functional on the dual X', that is, an element L of the bidual of X, and L is the limit of \{x_n\} in the weak*-topology of the bidual. The Banach space X is
weakly sequentially complete if every weakly Cauchy sequence is weakly convergent in X. It follows from the preceding discussion that reflexive spaces are weakly sequentially complete. An orthonormal sequence in a Hilbert space is a simple example of a weakly convergent sequence, with limit equal to the \mathbf{0} vector. The
unit vector basis of \ell^p for 1 or of c_0, is another example of a
weakly null sequence, that is, a sequence that converges weakly to \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 \mathbf{0}. The unit vector basis of \ell^1 is not weakly Cauchy. Weakly Cauchy sequences in \ell^1 are weakly convergent, since L^1-spaces are weakly sequentially complete. Actually, weakly convergent sequences in \ell^1 are norm convergent. This means that \ell^1 satisfies
Schur's property.
Results involving the basis Weakly Cauchy sequences and the \ell^1 basis are the opposite cases of the dichotomy established in the following deep result of
Haskell P. Rosenthal. {{math theorem| name = Theorem | math_statement = Let \{x_n\}_{n \in \N} be a bounded sequence in a Banach space. Either \{x_n\}_{n \in \N} has a weakly Cauchy subsequence, or it admits a subsequence
equivalent to the standard unit vector basis of \ell^1.}} A complement to this result is due to Odell and Rosenthal (1975). {{math theorem| name = Theorem | math_statement = Let X be a separable Banach space. The following are equivalent: • The space X does not contain a closed subspace isomorphic to \ell^1. • Every element of the bidual X'' is the weak*-limit of a sequence \{x_n\} in X.}} By the Goldstine theorem, every element of the unit ball B
of X is weak*-limit of a net in the unit ball of X. When X does not contain \ell^1, every element of B'' is weak*-limit of a in the unit ball of X. When the Banach space X is separable, the unit ball of the dual X', equipped with the weak*-topology, is a metrizable compact space K,
Sequences, weak and weak* compactness When X is separable, the unit ball of the dual is weak*-compact by the
Banach–Alaoglu theorem and metrizable for the weak* topology, If the dual X' is separable, the weak topology of the unit ball of 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. {{math theorem| name =
Eberlein–Šmulian theorem | math_statement = A set A in a Banach space is relatively weakly compact if and only if every sequence \{a_n\} in A has a weakly convergent subsequence.}} A Banach space X is reflexive if and only if each bounded sequence in X has a weakly convergent subsequence. A weakly compact subset A in \ell^1 is norm-compact. Indeed, every sequence in A has weakly convergent subsequences by Eberlein–Šmulian, that are norm convergent by the Schur property of \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==