1. <A,B>=<B,A> ;
2. <(A+B),C=<A,C>+<B,C> ;
3. <(kA),B=<A,(kB)>=<k,(A,B)> ;
4. <A,A> is greater than or equal to 0. <A,A> is 0 if and only if A=0.

1. Let f(x) and g(x) be two functions from C[0,1] (continuous functions on the interval [0,1]). Then f(x)g(x)=g(x)f(x) for every x in [0,1], so the integrals of f(x)g(x) and g(x)f(x) are equal. The first of these integrals is the dot product of f and g, the second one is the dot product of g and f. So

I leave properties 2 and 3 as exercises.

4. The dot product < f(x),f(x)> is the integral of f²(x) from 0 to 1. The function f²(x) is non-negative. Therefore the integral < f(x),f(x)> is non-negative. If f(x)=0 for every x then f²(x)=0 and the integral is 0. Conversely, if the integral of a non-negative continuous function is 0 then the function is 0 (a property of integrals), so if < f(x),f(x)> = 0 then f(x)=0. The proof is complete.

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