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Mathematics for Physics

218 Banach Space

A set of vectors with a norm where every sequence of elements that get closer to each other has a limit in the set that is used to study the continuity of a mapping.

definition [d] (Banach Space) From Kreyszig Definition 2.2-1: a Banach space is a complete normed space, complete in the metric defined by the norm

  • \(d(x,y) = \lVert x - y \rVert\) .

where

  • the underlying space is a normed space \((X, \lVert\cdot\rVert)\).
  • completeness means every Cauchy sequence in \(X\) converges to an element of \(X\).

218.1 Elementary Example

218.1.1 Simple

A Banach space is a complete normed space. The line \(\mathbb{R}\) with absolute value as norm is Banach.

\[ X = \mathbb{R},\quad \lVert x \rVert = |x| \]

\[ d(x,y) = |x - y| \]

where

  • \((X,\lVert\cdot\rVert)\) is a normed space.
  • completeness is with respect to \(d(x,y) = \lVert x - y \rVert\).

218.1.2 General

The space \(\mathbb{R}^{3}\) with the Euclidean norm is a Banach space.

\[ X = \mathbb{R}^{3},\quad \lVert x \rVert = \sqrt{x_{1}^{2} + x_{2}^{2} + x_{3}^{2}} \]

\[ d(x,y) = \lVert x - y \rVert \]

where

  • every Cauchy sequence in \(X\) converges to an element of \(X\).

218.2 References

  1. Kreyszig, E. Introductory Functional Analysis with Applications. Wiley, 1989. — Definition 2.2-1 (Banach space as complete normed space).