Banach space

In mathematics, more specifically in functional analysis, a Banach space (/ˈbɑː.nʌx/, Polish pronunciation: [ˈba.nax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space.

Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920-1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space". Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces.

A Banach space is a complete normed space \((X, \|{\cdot}\|).\) A normed space is a pair \((X, \|{\cdot}\|)\) consisting of a vector space \(X\) over a scalar field \(\mathbb{K}\) (where \(\mathbb{K}\) is commonly \(\Reals\) or \(\Complex\)) together with a distinguished norm \(\|{\cdot}\| : X \to \Reals.\) Like all norms, this norm induces a translation invariant distance function, called the canonical or (norm) induced metric, defined for all vectors \(x, y \in X\) by \[d(x, y) := \|y - x\| = \|x - y\|.\] This makes \(X\) into a metric space \((X, d).\) A sequence \(x_1, x_2, \ldots\) is called Cauchy in \((X, d)\) or \(d\)-Cauchy or \(\|{\cdot}\|\)-Cauchy if for every real \(r > 0,\) there exists some index \(N\) such that, for \(m\) and \(n\) are greater than \(N\) \[d(x_n, x_m) = \|x_n - x_m\| < r.\] The normed space \((X, \|{\cdot}\|)\) is called a Banach space and the canonical metric \(d\) is called a complete metric if \((X, d)\) is a complete metric space, which by definition means for every Cauchy sequence \(x_1, x_2, \ldots\) in \((X, d),\) there exists some \(x \in X\) such that \[\lim_{n \to \infty} x_n = x \; \text{ in } (X, d),\] where because \(\|x_n - x\| = d(x_n, x),\) this sequence's convergence to \(x\) can equivalently be expressed as \[\lim_{n \to \infty} \|x_n - x\| = 0 \; \text{ in } \Reals.\]

The norm \(\|{\cdot}\|\) of a normed space \((X, \|{\cdot}\|)\) is called a complete norm if \((X, \|{\cdot}\|)\) is a Banach space.

For any normed space \((X, \|{\cdot}\|),\) there exists an L-semi-inner product \(\langle\cdot, \cdot\rangle\) on \(X\) such that \(\|x\| = \sqrt{\langle x, x \rangle}\) for all \(x \in X.\) In general, there may be infinitely many L-semi-inner products that satisfy this condition and the proof of the existence of L-semi-inner products relies on the non-constructive Hahn-Banach theorem. L-semi-inner products are a generalization of inner products, which are what fundamentally distinguish Hilbert spaces from all other Banach spaces. This shows that all normed spaces (and hence all Banach spaces) can be considered as being generalizations of (pre-)Hilbert spaces.

The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. A normed space \(X\) is a Banach space if and only if each absolutely convergent series in \(X\) converges to a value that lies within \(X,\) symbolically \[\sum_{n=1}^{\infty} \|v_n\| < \infty \implies \sum_{n=1}^{\infty} v_n\text{ converges in } X.\]

The canonical metric \(d\) of a normed space \((X, \|{\cdot}\|)\) induces the usual metric topology \(\tau_d\) on \(X,\) which is referred to as the canonical or norm induced topology. Every normed space is automatically assumed to carry this Hausdorff topology, unless indicated otherwise. With this topology, every Banach space is a Baire space, although there exist normed spaces that are Baire but not Banach. The norm \(\|{\cdot}\| : X \to \Reals\) is always a continuous function with respect to the topology that it induces.

The open and closed balls of radius \(r > 0\) centered at a point \(x \in X\) are, respectively, the sets \[B_r(x) := \{z \in X \mid \|z - x\| < r\} \qquad \text{ and } \qquad C_r(x) := \{z \in X \mid \|z - x\| \leq r\}.\] Any such ball is a convex and bounded subset of \(X,\) but a compact ball/neighborhood exists if and only if \(X\) is finite-dimensional. In particular, no infinite, dimensional normed space can be locally compact or have the Heine-Borel property. If \(x_0\) is a vector and \(s \neq 0\) is a scalar, then \[x_0 + s\,B_r(x) = B_{|s| r}(x_0 + s x) \qquad \text{ and } \qquad x_0 + s\,C_r(x) = C_{|s| r}(x_0 + s x).\] Using \(s = 1\) shows that the norm-induced topology is translation invariant, which means that for any \(x \in X\) and \(S \subseteq X,\) the subset \(S\) is open (respectively, closed) in \(X\) if and only if its translation \(x + S := \{x + s \mid s \in S\}\) is open (respectively, closed). Consequently, the norm induced topology is completely determined by any neighbourhood basis at the origin. Some common neighborhood bases at the origin include \[\{B_r(0) \mid r > 0\}, \qquad \{C_r(0) \mid r > 0\}, \qquad \{B_{r_n}(0) \mid n \in \N\}, \qquad \text{ and } \qquad \{C_{r_n}(0) \mid n \in \N\},\] where \(r_1, r_2, \ldots\) can be any sequence of positive real numbers that converges to \(0\) in \(\R\) (common choices are \(r_n := \tfrac{1}{n}\) or \(r_n := 1/2^n\)). So, for example, any open subset \(U\) of \(X\) can be written as a union \[U = \bigcup_{x \in I} B_{r_x}(x) = \bigcup_{x \in I} x + B_{r_x}(0) = \bigcup_{x \in I} x + r_x\,B_1(0)\] indexed by some subset \(I \subseteq U,\) where each \(r_x\) may be chosen from the aforementioned sequence \(r_1, r_2, \ldots.\) (The open balls can also be replaced with closed balls, although the indexing set \(I\) and radii \(r_x\) may then also need to be replaced). Additionally, \(I\) can always be chosen to be countable if \(X\) is a separable space, which by definition means that \(X\) contains some countable dense subset.

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.

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). For finite (co)products, these norms give rise to isomorphic normed spaces, and the product \(X \times Y\) (or the direct sum \(X \oplus Y\)) is complete if and only if the two factors are complete.

If \(M\) is a closed linear subspace of a normed space \(X,\) there is a natural norm on the quotient space \(X / M,\) \[\|x + M\| = \inf\limits_{m \in M} \|x + m\|.\]

The quotient \(X / M\) is a Banach space when \(X\) is complete. 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 \[T = T_1 \circ \pi, \quad T : X \overset{\pi}{{}\longrightarrow{}} X/\ker T \overset{T_1}{{}\longrightarrow{}} Y\] where the first map \(\pi\) is the quotient map, and the second map \(T_1\) sends every class \(x + \ker T\) in the quotient to the image \(T(x)\) in \(Y.\) This is well defined because all elements in the same class have the same image. The mapping \(T_1\) is a linear bijection from \(X/\ker T\) onto the range \(T(X),\) whose inverse need not be bounded.

Basic examples 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.

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 \(a, b \in A.\)

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.

Hahn-Banach theorem, 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 orthogonal of \(M\) in the dual, \[M^{\bot} = \{ x' \in X \mid x'(m) = 0 \text{ for all } m \in M \}.\]

Theorem, Let \(X\) be a normed space. If \(X'\) is separable, then \(X\) is separable.

Condensed: the full section is in Wikipedia.

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 \(X\) 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 \(\sup_{T \in F} \|T(x)\|_Y < \infty,\) then \(\sup_{T \in F} \|T\|_Y < \infty.\)

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 \(U,\) \[\sup_{T \in F} \sup_{x \in U} \; \|T(x)\|_Y < \infty.\]

The Open Mapping Theorem, Let \(X\) and \(Y\) be Banach spaces and \(T : X \to Y\) be a surjective continuous linear operator, then \(T\) 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 \(X\) and \(Y\) are Banach spaces and that \(T \in B(X, Y).\) Suppose further that the range of \(T\) is closed in \(Y.\) Then \(X / \ker T\) is isomorphic to \(T(X).\)

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 \(X\) is the internal direct sum of closed subspaces \(M_1, \ldots, M_n,\) then \(X\) is isomorphic to \(M_1 \oplus \cdots \oplus M_n.\)

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 \(m_1 + \cdots + m_n.\)

The Closed Graph Theorem, Let \(T : X \to Y\) be a linear mapping between Banach spaces. The graph of \(T\) is closed in \(X \times Y\) if and only if \(T\) is continuous.

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.

Theorem, If \(X\) is a reflexive Banach space, every closed subspace of \(X\) and every quotient space of \(X\) 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 \(X\) onto the Banach space \(Y,\) then \(Y\) is reflexive.

Theorem, If \(X\) is a Banach space, then \(X\) is reflexive if and only if \(X'\) is reflexive.

Corollary, Let \(X\) be a reflexive Banach space. Then \(X\) is separable if and only if \(X'\) is separable.

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.

Theorem, Suppose that \(X_1, \ldots, X_n\) are normed spaces and that \(X = X_1 \oplus \cdots \oplus X_n.\) Then \(X\) is reflexive if and only if each \(X_j\) is reflexive.

Hilbert spaces are reflexive. The \(L^p\) spaces are reflexive when \(1 < p < \infty.\) 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.

Theorem, A Banach space \(X\) is reflexive if and only if its unit ball is compact in the weak topology.

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.

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

• \(X\) is reflexive.
• for all \(f\) in \(X'\) there exists \(x \in X\) with \(\|x\| \leq 1,\) so that \(f(x) = \|f\|.\)

Condensed: the full section is in Wikipedia.

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.

Theorem , For every measure \(\mu,\) the space \(L^1(\mu)\) 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 \(\mathbf{0}\) vector. The unit vector basis of \(\ell^p\) for \(1 < p < \infty,\) 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.

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.