Banach space

In mathematics, Banach spaces (pronounced [ˈbanax]) is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence (with respect to the metric d(x, y) = ||xy||) in V has a limit in V (with respect to the topology induced by that metric). As for general vector spaces, a Banach space over the real numbers is called a real Banach space, and a Banach space over the complex numbers is called a complex Banach space.

Many of the infinite-dimensional function spaces studied in analysis are Banach spaces, including spaces of continuous functions (continuous functions on a compact Hausdorff space), spaces of Lebesgue integrable functions known as Lp spaces, and spaces of holomorphic functions known as Hardy spaces. They are the most commonly used topological vector spaces, and their topology comes from a norm.

They are named after the Polish mathematician Stefan Banach, who introduced them in 1920–1922 along with Hans Hahn and Eduard Helly.[1]

Contents

Examples

Throughout, let K stand for one of the fields R or C.

The familiar Euclidean spaces Kn, where the Euclidean norm of x = (x1, …, xn) is given by ||x|| = (x12+…+ xn2)1/2, are Banach spaces. Hence every finite-dimensional K vector space becomes a Banach space being endowed with an arbitrary norm, since all norms are equivalent on a finite-dimensional K vector space.

Consider the space of all continuous functions ƒ : [ab] → K defined on a closed interval [ab]. This space becomes a Banach space if an appropriate norm ||ƒ||, is defined in it. Such a norm may be defined as ||ƒ|| = sup { |ƒ(x)| : x ∈ [ab] }, known as the supremum norm. This is indeed a well-defined norm, since continuous functions defined on a closed interval are bounded.

Since ƒ is a continuous function on a closed interval, then it is bounded and the supremum in the above definition is attained by the Weierstrass extreme value theorem, so we can replace the supremum by the maximum. In this case, the norm is also called the maximum norm.

The space is complete under this norm, and the resulting Banach space is denoted by C[ab]. This example can be generalized to the space C(X) of all continuous functions X → K, where X is a compact space, or to the space of all bounded continuous functions X → K, where X is any topological space, or indeed to the space B(X) of all bounded functions X → K, where X is any set. In all these examples, we can even multiply functions and stay in the same space: all these examples are in fact unital Banach algebras.

For any open set Ω ⊆ C, the set A(Ω) of all bounded, analytic functions u : Ω → C is a complex Banach space with respect to the supremum norm. The fact that uniform limits of analytic functions are analytic is an easy consequence of Morera's theorem.

If p ≥ 0 is a real number, we can consider the space of all infinite sequences (x1, x2, x3, …) of elements in K such that the infinite series ∑i |xi|p is finite. The p-th root of this series' value is then defined to be the p-norm of the sequence. The space, together with this norm, is a Banach space; it is denoted by ℓ p.

The Banach space ℓ consists of all bounded sequences of elements in K; the norm of such a sequence is defined to be the supremum of the absolute values of the sequence's members.

Again, if p ≥ 1 is a real number, we can consider all functions ƒ : [ab] → K such that |ƒ|p is Lebesgue integrable. The p-th root of this integral is then defined to be the norm of ƒ. By itself, this space is not a Banach space because there are non-zero functions whose norm is zero. We define an equivalence relation as follows: ƒ and g are equivalent if and only if the norm of ƒg is zero. The set of equivalence classes then forms a Banach space; it is denoted by Lp([ab]). It is crucial to use the Lebesgue integral and not the Riemann integral here, because the Riemann integral would not yield a complete space. These examples can be generalized; see Lp spaces for details.

If X and Y are two Banach spaces, then we can form their direct sum X ⊕ Y, which has a natural topological vector space structure but no canonical norm. However, it is again a Banach space for several equivalent norms, for example

 \|x \oplus y\| = \bigl( \|x\|^p + \|y\|^p \bigr)^{1/ p}, \quad 1 \le p \le \infty.

This construction can be generalized to define ℓp-direct sums of arbitrarily many Banach spaces. When there is an infinite number of non-zero summands, the space obtained in this way depends upon p.

If M is a closed linear subspace of the Banach space X, then the quotient space X / M is again a Banach space.

Every inner product gives rise to an associated norm. The inner product space is called a Hilbert space if its associated norm is complete. Thus every Hilbert space is a Banach space by definition. The converse statement also holds under certain conditions; see below.

Linear operators

If V and W are Banach spaces over the same ground field K, the set of all continuous K-linear maps A : VW is denoted by L(V, W). 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 L(V, W) can be given the operator norm

\|A\| = \sup\{\|Ax\|_W \mid x\in V,\ \|x\|_V\le 1\}.

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

When V = W, the space L(V) = L(V, V) forms a unital Banach algebra; the multiplication operation is given by the composition of linear maps.

Dual space

If V is a Banach 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 dual space V′ as V′ = L(V, K), the space of continuous linear maps into K. This is again a Banach space (with the operator norm). It can be used to define a new topology on V: the weak topology.

Note that the requirement that the maps be continuous is essential; if V is infinite-dimensional, there exist linear maps which are not continuous, and therefore not bounded, so the space V of linear maps into K is not a Banach space. The space V (which may be called the algebraic dual space to distinguish it from V') also induces a weak topology which is finer than that induced by the continuous dual since V ′ ⊆ V.

There is a natural map F  from V to V′′ (the dual of the dual) defined by

F(x)(ƒ) = ƒ(x)

for all x in V and ƒ in V′. Because F(x) is a map from V′ to K, it is an element of V′′. The map F: x → F(x) is thus a map V → V′′. As a consequence of the Hahn–Banach theorem, this map is injective, and isometric; if it is also surjective, then the Banach space V is called reflexive. Reflexive spaces have many important geometric properties. A space is reflexive if and only if its dual is reflexive, which is the case if and only if its unit ball is compact in the weak topology.

For example, ℓp is reflexive for 1 < p < ∞ but ℓ1 and ℓ are not reflexive. When p < ∞, the dual of ℓp is ℓq where p and q are related by the formula 1/p + 1/q = 1. See L p spaces for details.

Relationship to Hilbert spaces

As mentioned above, every Hilbert space 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 ||v||² = (v,v) for all v.

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

\|u+v\|^2 + \|u-v\|^2 = 2(\|u\|^2 + \|v\|^2)

for all u and v in V, and where ||*|| is the norm on V. 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 Lp 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 V is a real Banach space, then the polarization identity is

\langle u,v\rangle = \frac{1}{4} (\|u+v\|^2 - \|u-v\|^2)

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

\langle u,v\rangle = \frac{1}{4} \left(\|u+v\|^2 - \|u-v\|^2 + i(\|u+iv\|^2 - \|u-iv\|^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 check also that the bilinear form is linear over i in one argument, and conjugate linear in the other.

Hamel dimension

It follows from the completeness of Banach spaces and the Baire category theorem that a Hamel basis of an infinite-dimensional Banach space is uncountable.

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 RR 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

Notes

  1. ^ Bourbaki 1987, V.86

References


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