freeCodeCamp/guide/english/mathematics/linear-algebra/inner-product-spaces/index.md

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Inner Product Spaces

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