# Hilbert space

From Maths

## Definition

A *Hilbert space*^{[1]} is a vector space, [ilmath](H,F)[/ilmath] (where [ilmath]F[/ilmath] is either [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath]) with an Inner product [math]\langle\cdot,\cdot\rangle[/math] such that [ilmath]H[/ilmath] is complete with respect to the associated norm [math]\|x\|=\sqrt{\langle x,x\rangle}[/math]

That is to say a Hilbert space is a Banach space where the norm is given by an inner product

## References

- ↑ Functional Analysis I - Lecture Notes - Richard Sharp - Sep 2014