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title: Inner Product Spaces
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## Inner Product Spaces

### Introduction

Let V be a vector space over field F. An inner product is a function that assigns to every ordered pair of vector x and y in V, a scalar in F, denoted by <x,y> such that for all x,y in V and a in F these hold:

* <x+z,y>=<x,y>+<x,z>
* <ax,y>=a<x,y>
* <X,Y>=<y,x> (X and Y denote conjugate of x and y respectively)
* <x,x>=0 for all x!=0
