Let \(V\) be a vector space over \(F\in \{\mathbb{R}, \mathbb{C}\}\). An inner product is a bilinear map \(\langle \cdot, \cdot\rangle\colon V\times V\to F\) which is conjugate symmetric and positive definite, i.e. \(\langle x, x\rangle>0\) if \(x\neq 0\).
Remark
- The Cauchy-Schwartz inequality implies continuity of the inner product as a map from \(V\times V\) to \(\mathbb{R}\), where \(V\times V\) is endowed with the norm \[ \lVert (u,v)\rVert_{V\times V} := (\lVert u\rVert^2_V + \lVert v\rVert^2_V)^{1/2} \] for \(u,v\in V\).
Connection to norms Link to heading
- \(\lVert v\rVert=\sqrt{\langle v, v\rangle}\) defines a norm .
- An inner product satisfies the polarization identity .
- A norm defines an inner product via the polarization identity if it satisfies the parallelogram law (see (0x68e4c8b2) ).