One of the most common classes of vector space encountered in mathematics, especially functional analysis.

Definition: A Banach space is a normed vector space which is complete in the metric induced by its norm.

That is, start with a vector space E over the real numbers R or complex numbers C (or you could use another complete normed field such as the p-adic numbers, but that's a whole 'nother can of worms). Call your field of choice k. Define a norm || || on E, which is a notion of absolute value or distance from zero that is compatible with the vector space structure on E:

  • ||x|| is a nonnegative real number for all x ∈ E, and ||x|| = 0 if and only if x = 0.
  • if α ∈ k (α is a number) and x ∈ E, then ||αx|| = |α| ||x||, where |α| is measured using the usual absolute value on k.
  • if x, y in E, then ||x + y|| ≤ ||x|| + ||y||. This is the familiar triangle inequality for absolute value.
If you think of ||x|| as the distance of x from zero, then || || defines a metric d on E, by setting d(x, y) = ||x - y||. This turns E into a metric space. If E is complete under this metric, then we call E a Banach space.

Every finite-dimensional vector space over the real or complex numbers is a Banach space, using the conventional Euclidean norm

||(x1, ..., xn)|| = (|x1|2 + ... + |xn|2)1/2.

(A perverse way to describe this norm on Rn would be to call it L2R{1, ..., n}. See below.) In fact, it is not hard to prove that every finite-dimensional Banach space is isomorphic to one of these. The interest of Banach spaces lies in the many nontrivial infinite-dimensional examples. These include:

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