Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Informally, 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 in the space.

Banach spaces are named after the Polish mathematician Stefan Banach, who introduced and made a systematic study of them in 1920–1922 along with Hans Hahn and Eduard Helly. Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Reisz 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.

Definition

A Banach space is a vector space X over the field of real numbers R or complex numbers C which is equipped with a norm and which is complete with respect to that norm. Formally, the definition of a Banach space is :

A normed space X is said to be a Banach space if for every Cauchy sequence $\left\{x_{n}\right\}_{n=1}^{\infty }\subset X$ there exists an element x in X such that $\lim _{n\to \infty }x_{n}=x$.

Explanations

In metric spaces, the completeness is a property of the metric. It is not a property of the topological space itself. If you move on to an equivalent metric (that is a metric which induces the same topology), the completeness can get lost. Regarding two equivalent norms on a normed vector space, however, one of them is complete if and only if the other one is complete. Therefore, in the case of normed vector spaces, the completeness is a property of the norm topology, which does not depend on the specific norm.

Theorems and Properties

• Theorem A normed space X is a Banach space if and only if each absolutely convergent series in X converges.

• Theorem Let X be a normed space. Then there is a Banach space Y and an isometric isomorphism T: X → Y such that T(X) is dense in Y. Furthermore, the space X′ is isometrically isomorphic to Y′. If Z is another Banach space such that there is an isometric isomorphism from X onto a dense subset of Z, then Z is isometrically isomorphic to Y.

• Theorem Let X and Y be normed spaces. Then, B(X, Y) := {T: X → Y | T linear and bounded}, is a normed space under the operator norm. If Y is a Banach space, then so is B(X, Y).

• Proposition Let T be a linear operator from a normed space X into a normed space Y. If X is a Banach space and T is an isomorphism, then T(X) is a Banach space.

• Corollary Every finite-dimensional normed space is a Banach space.

• Corollary A Banach space with a countable Hamel basis is finite-dimensional.

• The Open Mapping Theorem Let X and Y be Banach spaces and T: X → Y be a continuous linear operator. Then T is surjective if and only if T is an open map. In particular, if T is bijective and continuous, then T⁻¹ is also continuous.

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

• The Closed Graph Theorem Let T: X → Y be a linear function between Banach spaces. The graph is closed in X × Y if and only if T is continuous.

• Theorem If M is a closed subspace of a Banach space X, then X/M with $\|x+M\|=\inf \limits _{m\in M}\|x+m\|$ is also a Banach space.

• The First Isomorphism theorem for Banach spaces Suppose that X and Y are Banach spaces and that T ∈ B(X, Y). Suppose further that the range of T is closed in Y. Then X/Ker(T) ≅ T(X). This is a topological isomorphism with means that a bijective, linear set L exists which goes from X/Ker(T) to T(X) so that both L and L⁻¹ are continuous.
• Theorem Let X₁, ..., X_n be normed spaces. Then X₁ ⊕ ... ⊕ X_n is a Banach space if and only if each X_j is a Banach space.
• Proposition If X is a Banach space that is the internal direct sum of its closed subspaces M₁, ..., M_n, then X ≅ M₁ ⊕ ... ⊕ M_n.
• Theorem Every Banach space is a Fréchet space.
• Theorem For every separable Banach space X, there is a closed subspace M of ℓ₁ such that X ≅ ℓ₁/M.
• Hahn–Banach theorem Let X be a vector space of the field K. Let further
  • Y ⊆ X be a linear subspace,
  • p: X → R be a sublinear function and
  • f: Y → K be a linear functional so that Re f(y) ≤ p(y) for all y in Y.
    Then, there exists a linear functional F: X → K so that
  • $F|_{Y}=f\,\,$ and
  • $\operatorname {Re} F(x)\leq p(x)\forall x\in X$.

In particular, every continuous linear functional on a subspace of a normed space can continuously be continued on the whole space.

• 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 $\sup _{T\in F}\|T(x)\|<\infty$, then $\ \sup _{T\in F}\|T\|<\infty .$
• Banach–Alaoglu theorem Let X be a normed vector space. Then the closed unit ball of the dual space B′ := {x ∈ X′ | ǁxǁ ≤ 1} is compact in the weak* topology.
• Theorem If there are Banach spaces which are invariant under the action of an integrable group representation and give their atomic decompositions with respect to coherent states, then the atoms arise from a single element under the group action.

• Theorem If X is a separable infinite dimensional Banach space then its isomorphism class has infinite diameter with respect to the Banach-Mazur distance.

Linear operators

Main article: Bounded operator

If X and Y are Banach spaces over the same ground field K, the set of all continuous K-linear maps T: X → Y is denoted by B(X, Y). In infinite-dimensional spaces, not all linear maps are automatically continuous. In general, a linear mapping on a normed space is continuous if and only if it is bounded on the closed unit ball. Thus, the vector space B(X, Y) can be given the operator norm

$\|T\|=\sup\{\|Tx\|_{Y}\mid x\in X,\ \|x\|_{X}\leq 1\}.$

With respect to this norm B(X, Y) is a Banach space. This is also true under the less restrictive condition that X be a normed space.

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.

Dual space

Main article: Dual space

If X is a normed space and K is the underlying field (either the real or the complex numbers), then K is itself a Banach space (using the absolute value as norm) and we can define the continuous dual space as X′ = B(X, K), the space of continuous linear maps from X into K.

• Theorem If X is a normed space, then X′ is a Banach space.
• Theorem Let X be a normed space. If X′ is separable, then X is separable.

The continuous dual space can be used to define a new topology on X: the weak topology. Note that the requirement that the maps be continuous is essential; if X is infinite-dimensional, there exist linear maps which are not continuous, and therefore not bounded. The space X* of all linear maps into K (which is called the algebraic dual space to distinguish it from X′) also induces a weak topology which is finer than that induced by the continuous dual since X′ ⊆ X*.

There is a natural map F: X → X′′ (the dual of the dual = bidual) defined by

F(x)(f) = f(x) for all x in X and f in X′.

Because F(x) is a map from X′ to K, it is an element of X′′. The map F: x → F(x) is thus a map X → X′′. As a consequence of the Hahn–Banach theorem, this map is injective, and isometric.

Reflexivity

If F is also surjective, then the Banach space X is called reflexive. Reflexive spaces have many important geometric properties.

• Theorem Every reflexive normed space is a Banach space.
• Corollary If X is a Banach space, then X is reflexive if and only if X′ is reflexive, which is the case if and only if its unit ball is compact in the weak topology.
• Corollary Suppose that X₁, ..., X_n are normed spaces and that X = X₁ ⊕ ... ⊕ X_n. Then X is reflexive if and only if each X_j is reflexive.
• Corollary Let X be a reflexive normed space and Y a Banach space. If there is a bounded linear operator from X onto Y, then Y is reflexive.
• Corollary Let X be a reflexive normed space. Then X is separable if and only if X′ is separable.
• James`s 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 ǁxǁ ≤ 1, so that f(x) = ǁfǁ.
• Lemma A Banach space X is reflexive if and only if the natural pairing on X × X′ is perfect. In particular, X′ is also reflexive then.

Tensor product

Main article: Tensor product

Let X and Y be two K-vector spaces. The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T′: X × Y → Z′ is any bilinear function into a K-vector space Z′, then only one linear function f: Z → Z′ with $T'=f\circ T$ exists.

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. In general, the tensor product of complete spaces is not complete again.

Placement in the hierarchy of mathematical structures

Overview over abstract spaces. An arrow must be understood as an implication. That means a space at the beginning of an arrow is also a space at the end of an arrow.

Every Hilbert space X is a Banach space because, by definition, a Hilbert space is complete with respect to the norm associated with its inner product, where a norm and an inner product are said to be associated if $\|x\|^{2}=\langle x,x\rangle$ for all x ∈ X.

The converse is not always true; not every Banach space is a Hilbert space. A necessary and sufficient condition for a Banach space X to be associated to an inner product (which will then necessarily make X into a Hilbert space) is the parallelogram identity:

$\|x+y\|^{2}+\|x-y\|^{2}=2(\|x\|^{2}+\|y\|^{2})$

for all x and y in X, and where $\|\cdot \|$ is the norm on X. So, for example, while Rⁿ is a Banach space with respect to any norm defined on it, it is only a Hilbert space with respect to the Euclidean norm. Similarly, as an infinite-dimensional example, the Lebesgue space L^p is always a Banach space but is only a Hilbert space when p = 2.

If the norm of a Banach space satisfies this identity, the associated inner product which makes it into a Hilbert space is given by the polarization identity. If X is a real Banach space, then the polarization identity is

$\langle x,y\rangle ={\frac {1}{4}}(\|x+y\|^{2}-\|x-y\|^{2})$

whereas if X is a complex Banach space, then the polarization identity is given by (assuming that scalar product is linear in first argument):

$\langle x,y\rangle ={\frac {1}{4}}\left(\|x+y\|^{2}-\|x-y\|^{2}+i(\|x+iy\|^{2}-\|x-iy\|^{2})\right).$

The necessity of this condition follows easily from the properties of an inner product. To see that it is sufficient—that the parallelogram law implies that the form defined by the polarization identity is indeed a complete inner product—one verifies algebraically that this form is additive, whence it follows by induction that the form is linear over the integers and rationals. Then, since every real is the limit of some Cauchy sequence of rationals, the completeness of the norm extends the linearity to the whole real line. In the complex case, one can also check that the bilinear form is linear over i in one argument, and conjugate linear in the other.

Examples

Main article: List of Banach spaces

Here K denotes the field of real numbers or complex numbers, I is a closed and bounded interval [a, b] and p, q are real numbers with 1 < p, q < ∞ so that

${\frac {1}{q}}+{\frac {1}{p}}=1$.

The symbol Σ denotes a σ-algebra of sets, and Ξ denotes just an algebra of sets (for spaces only requiring finite additivity, such as the ba space). The symbol μ denotes a positive measure: that is, a real-valued positive set function defined on a σ-algebra which is countably additive.

Classical Banach spaces
	Dual space	Reflexive	weakly complete	Norm	Notes
Kn	Kn	Yes	Yes	$\|x\|_{2}=\left(\sum _{i=1}^{n}|x_{i}|^{2}\right)^{1/2}$
ℓnp	ℓnq	Yes	Yes	$\|x\|_{p}=\left(\sum _{i=1}^{n}|x_{i}|^{p}\right)^{1/p}$
ℓn∞	ℓn1	Yes	Yes	$\|x\|_{\infty }=\max _{1\leq i\leq n}|x_{i}|$
ℓp	ℓq	Yes	Yes	$\|x\|_{p}=\left(\sum _{i=1}^{\infty }|x_{i}|^{p}\right)^{1/p}$
ℓ1	ℓ∞	No	Yes	$\|x\|_{1}=\sum _{i=1}^{\infty }|x_{i}|$
ℓ∞	ba	No	No	$\|x\|_{\infty }=\sup _{i}|x_{i}|$
c	ℓ1	No	No	$\|x\|_{\infty }=\sup _{i}|x_{i}|$
c0	ℓ1	No	No	$\|x\|_{\infty }=\sup _{i}|x_{i}|$	Isomorphic but not isometric to c.
bv	ℓ1 + K	No	Yes	$\|x\|_{bv}=|x_{1}|+\sum _{i=1}^{\infty }|x_{i+1}-x_{i}|$
bv0	ℓ1	No	Yes	$\|x\|_{bv_{0}}=\sum _{i=1}^{\infty }|x_{i+1}-x_{i}|$
bs	ba	No	No	$\|x\|_{bs}=\sup _{n}\left|\sum _{i=1}^{n}x_{i}\right|$	Isometrically isomorphic to ℓ∞.
cs	ℓ1	No	No	$\|x\|_{bs}=\sup _{n}\left|\sum _{i=1}^{n}x_{i}\right|$	Isometrically isomorphic to c.
B(X, Ξ)	ba(Ξ)	No	No	$\|f\|_{B}=\sup _{x\in X}|f(x)|$
C(X)	rca(X)	No	No	$\|x\|_{bs}=\sup _{x\in X}\left|f(x)\right|$	X is a compact Hausdorff space.
ba(Ξ)	?	No	Yes	$\|\mu \|_{ba}=\sup _{A\in \Sigma }|\mu |(A)$ (variation of a measure)
ca(Σ)	?	No	Yes	$\|\mu \|_{ba}=\sup _{A\in \Sigma }|\mu |(A)$	A closed subspace of ba(Σ).
rca(Σ)	?	No	Yes	$\|\mu \|_{ba}=\sup _{A\in \Sigma }|\mu |(A)$	A closed subspace of ca(Σ).
Lp(μ)	Lq(μ)	Yes	Yes	$\|f\|_{p}=\left\{\int |f|^{p}\,d\mu \right\}^{1/p}$
BV(I)	?	No	Yes	$\|f\|_{BV}=\lim _{x\to a^{+}}f(x)+V_{f}(I)$	Vf(I) is the total variation of f.
NBV(I)	?	No	Yes	$\|f\|_{BV}=V_{f}(I)$	NBV(I) consists of BV functions such that $\lim _{x\to a^{+}}f(x)=0$.
AC(I)	K+L∞(I)	No	Yes	$\|f\|_{BV}=\lim _{x\to a^{+}}f(x)+V_{f}(I)$	Isomorphic to the Sobolev space W1,1(I).
Cn[a,b]	rca([a,b])	No	No	$\|f\|=\sum _{i=0}^{n}\sup _{x\in [a,b]}|f^{(i)}(x)|.$	Isomorphic to Rn ⊕ C([a,b]), essentially by Taylor's theorem.

Derivatives

Several concepts of a derivative may be defined on a Banach space. See the articles on the Fréchet derivative and the Gâteaux derivative.

Generalizations

Several important spaces in functional analysis, for instance the space of all infinitely often differentiable functions R → R or the space of all distributions on R, are complete but are not normed vector spaces and hence not Banach spaces. In Fréchet spaces one still has a complete metric, while LF-spaces are complete uniform vector spaces arising as limits of Fréchet spaces.

See also

• Space (mathematics)
  • Hilbert space
• Distortion problem

References

• Banach, Stefan (1932), Théorie des opérations linéaires, Monografie Matematyczne, vol. 1, Warszawa: Subwencji Funduszu Kultury Narodowej, Zbl 0005.20901.
• Beauzamy, Bernard (1985 [1982]), Introduction to Banach Spaces and their Geometry (Second revised ed.), North-Holland {{citation}}: Check date values in: |year= (help)CS1 maint: year (link).
• Bourbaki, Nicolas (1987), Topological vector spaces, Elements of mathematics, Berlin: Springer-Verlag, ISBN 978-3-540-13627-9.
• Dunford, Nelson; Schwartz, Jacob T. (1958), Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York, MR 0117523
• Prof. Dr. A. Deitmar: Funktionalanalysis Skript WS2011/12 < http://www.mathematik.uni-tuebingen.de/~deitmar/LEHRE/frueher/2011-12/FA/FA.pdf>
• Megginson, Robert E. (1991), An Introduction to Banach Space Theory, Springer-Verlag.
• Ryan, Raymond A. (2000), Introduction to Tensor Products of Banach Spaces, Springer-Verlag.
• Willkomm, Anton (1976), Über die Darstellungstheorie topologischer Gruppen in nicht-archimedischen Banach-Räumen, Dissertation, Rheinisch-Wetfälische Technische Hochschule Aachen.