231 Normed Space
A set of vectors with a norm defined on it that is used to measure distance between vectors by the metric induced by the norm.
Note: Also called normed vector space. Also called normed linear space.
definition [d] (Normed Space) From Kreyszig Definition 2.2-1: a normed space \(X\) is a vector space with a norm defined on it. A norm on a real or complex vector space \(X\) is a real-valued function on \(X\) whose value at an \(x \in X\) is denoted by \(\lVert x \rVert\) and which has the properties
- (N1) \(\lVert x \rVert \geq 0\)
- (N2) \(\lVert x \rVert = 0\) if and only if \(x = 0\)
- (N3) \(\lVert \alpha x \rVert = |\alpha|\, \lVert x \rVert\)
- (N4) \(\lVert x + y \rVert \leq \lVert x \rVert + \lVert y \rVert\)
A norm on \(X\) defines a metric \(d\) on \(X\) by
- \(d(x,y) = \lVert x - y \rVert\)
called the metric induced by the norm. The normed space is denoted \((X, \lVert\cdot\rVert)\) or simply \(X\).
where
- \(X\) is a real or complex vector space.
- \(\lVert\cdot\rVert\) is the norm.
- \(d\) is the metric induced by the norm.
231.1 Elementary Example
231.1.1 Simple
A normed space is a vector space with a norm. Take \(\mathbb{R}^{2}\) with the Euclidean norm.
\[ X = \mathbb{R}^{2},\quad \lVert (x_{1},x_{2}) \rVert = \sqrt{x_{1}^{2} + x_{2}^{2}} \]
\[ d(x,y) = \lVert x - y \rVert \]
where
- \((X,\lVert\cdot\rVert)\) is the normed space.
- \(d\) is the metric induced by the norm.
231.1.2 General
The space \(\mathbb{R}^{3}\) with the Euclidean norm is a normed space whose induced metric is the usual distance in three dimensions.
\[ X = \mathbb{R}^{3},\quad \lVert x \rVert = \sqrt{x_{1}^{2} + x_{2}^{2} + x_{3}^{2}} \]
\[ d(x,y) = \lVert x - y \rVert \]
where
- \(x = (x_{1},x_{2},x_{3})\) is a vector in \(X\).