Normed Space

Author

John Robin Inston

Published

September 25, 2026

1 Norm

For a vector space \(V\) over \(\mathbb{R}\), a norm is a function \(\lVert\cdot\rVert:V\to\mathbb{R}\) satisfying, for all \(v,w\in V\) and \(\alpha\in\mathbb{R}\):

  1. \(\lVert v\rVert\geq0\);
  2. \(\lVert v\rVert=0\) if and only if \(v=0\);
  3. \(\lVert\alpha v\rVert=\lvert\alpha\rvert\lVert v\rVert\) (homogeneity);
  4. \(\lVert v+w\rVert\leq\lVert v\rVert+\lVert w\rVert\) (triangle inequality).

2 Normed Space

The pair \((V,\lVert\cdot\rVert)\) of a vector space and a norm is a normed space. A Banach space is a normed space that is complete, that is, every Cauchy sequence converges (see Metric Spaces).

If \(\lVert\cdot\rVert\) is a norm on \(V\) then \(d(v,w):=\lVert v-w\rVert\) is a metric on \(V\).

Proof. Positivity and \(d(v,w)=0\iff v=w\) follow from norm axioms 1 and 2. Symmetry follows from axiom 3 with \(\alpha=-1\). The triangle inequality for \(d\) follows from axiom 4 applied to \(v-u=(v-w)+(w-u)\).

3 Examples

  • \(\mathbb{R}^n\) with the \(p\)-norm \(\lVert x\rVert_p=\left(\sum_i\lvert x_i\rvert^p\right)^{1/p}\) for \(p\geq1\), see Lp Norm.
  • \(C([0,1])\) with the sup norm \(\lVert f\rVert_\infty=\sup_{x}\lvert f(x)\rvert\), which is a Banach space, and with the integral norm \(\lVert f\rVert_1=\int_0^1\lvert f\rvert\), which is not complete.

An inner product always induces a norm \(\lVert v\rVert=\sqrt{\langle v,v\rangle}\); a complete inner product space is a Hilbert space.

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