230 Norm
A norm is a function that associates a real number with a vector, that is used for measuring the distance between vectors.
definition [d] (Norm) From Kreyszig: 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) \(\lVert x + y \rVert \leq \lVert x \rVert + \lVert y \rVert\) (Triangle inequality)
here \(x\) and \(y\) are arbitrary vectors in \(X\) and \(\alpha\) is any scalar.
where
- \(X\) is a real or complex vector space.
- \(\lVert x \rVert\) is the norm of \(x\).
- \(\alpha\) is a scalar.
- \(|\alpha|\) is the absolute value of \(\alpha\).
Note:
- Kreyszig: 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 \(X\).
230.1 Elementary Example
230.1.1 Simple
A norm assigns a nonnegative length to each vector. On \(\mathbb{R}^{2}\) use the Euclidean length.
\[ X = \mathbb{R}^{2} \]
\[ \lVert (x_{1},x_{2}) \rVert = \sqrt{x_{1}^{2} + x_{2}^{2}} \]
\[ \lVert (3,4) \rVert = 5,\quad \lVert (0,0) \rVert = 0 \]
where
- \(\lVert x \rVert\) is the norm of the vector \(x\).
- \(X\) is the vector space.
230.1.2 General
On \(\mathbb{R}^{3}\) the same Euclidean norm has three squared components, and it induces the metric \(d(x,y) = \lVert x - y \rVert\).
\[ \lVert (x_{1},x_{2},x_{3}) \rVert = \sqrt{x_{1}^{2} + x_{2}^{2} + x_{3}^{2}} \]
\[ d(x,y) = \lVert x - y \rVert \]
where
- \(d\) is the metric induced by the norm.