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Mathematics for Physics

354 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 \(X\).

354.1 Examples

354.1.1 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.

354.1.2 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.

354.2 References

  1. Kreyszig, E. Introductory Functional Analysis with Applications. Wiley, 1989. — Norm axioms (N1)–(N4); metric induced by the norm \(d(x,y)=\lVert x-y\rVert\).