Inner product space

Definition

An inner product space (AKA an i.p.s or a pre-hilbert space) is a:

• Vector space (over the field $\mathbb{R}$ or $\mathbb{C}$, which we shall denote $F$) $(X,F)$, equipped with an
• Inner product, $\langle\cdot,\cdot\rangle$

We denote this $(X,\langle\cdot,\cdot\rangle,F)$ or just $(X,\langle\cdot,\cdot\rangle)$ if the field is implicit.

Notes

• See the article Subtypes of topological spaces for more information.

All i.p.s are also normed spaces as there is an induced norm on an i.p.s given by:

• For an $x\in X$ we define $$\Vert x\Vert:=\sqrt{\langle x,x\rangle^2}$$

(which as per the article in turn induces its own metric: $d(x,y):=\Vert x-y\Vert$)