# Inner product space

## Definition

An inner product space (AKA an i.p.s or a pre-hilbert space) is a[1]:

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

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

## Notes

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

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

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

## References

1. Functional Analysis - George Bachman and Lawrence Narici