Friedberg's Linear Algebra states in one of the exercises that an inner product space must be over the field of real or complex numbers. After looking at the definition for while, I am still having trouble seeing why this must be so. The definition of a inner product space is given as follows.(adsbygoogle = window.adsbygoogle || []).push({});

Let V be a vector space over F. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F, denoted <x,y>, such that for all x, y, and z in V and all c in F, the following hold:

(a) <x+z, y>=<x,y> + <z,y>

(b) <cx,y>=c<x,y>

(c) <x,y>=[itex]\overline{<y,x>}[/itex]

(d) <x,x> > 0 if x[itex]\neq[/itex]0.

I can't convince myself that I could not contrive some vector space and some inner product such that the resulting inner product space would not use the whole number line.

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# Why must inner product spaces be over the field of real or complex numbers?

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