Banach space

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Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. Banach spaces are typically infinite-dimensional spaces containing functions.

Mathematically correct, Banach spaces are defined as complete normed vector spaces. This means that a Banach space is a vector space V with a norm ||.|| such that every Cauchy sequence in V converges. See metric space for a discussion of the concept of completeness.

Every Hilbert space is a Banach space, but there are several other important examples. Consider for instance the space of all continuous functions f : [a, b] -> R defined on a closed interval [a, b] with values in the real numbers R. The norm ||f|| of such a function can be defined as ||f|| = sup { |f(x)| : x in [a, b] }. This is indeed a norm since continuous functions defined on a closed interval are bounded. The space is complete under this norm, and the resulting Banach space is denoted by C[a, b]. Another commonly used Banach space is the space of all sequences (an) such that the series ∑ |an| converges; this sum is then defined to be the norm of the sequence. The space is complete under this norm and is denoted by l1.


If V and W are Banach spaces, the set of all continuous linear maps A : V -> W is denoted by L(V, W). This is a vector space, and by defining the norm ||A|| = sup { ||Ax|| : x in V with ||x|| ≤ 1 } it can be turned into a Banach 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 and we can define the dual space V' by V' = L(V, K). This is again a Banach space. There is a natural map Φ from V to V'' defined by

Φ(x)(φ) = φ(x)

for all φ in V'. As a consequence of the Hahn-Banach theorem, this map is injective; 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.