(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider the linear space S, which consists of square integrable continuous

functions in [0,1]. These are continuous functions x : [0,1] -> R such that the integral is less than infinity.

2. Relevant equations

Show that the operation

∫x(t)y(y)dt at [0,1] is an inner product in this space.

3. The attempt at a solution

Apparently there are three axioms:

1] <x,x> is >= 0 (<x,x> = 0 only if x is a trivial function). I think that know how to prove this one. Simply, if the integrand is positive at any point then the function must be positive in the neighborhood of that point. Therefore, the integral cannot be <= 0. It can be zero only if the function is trivial such that x(t) = 0.

2] Linearity - how would I prove the linearity argument for general continuous functions? Could I replace such function with its Taylor series and show the linearity on the resulting polynomials?

3] Symmetry - no clue so far.

Could you guys give me some pointers?

Thanks a lot,

SunnyBoy

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# Homework Help: Inner Product Proof on Square Integrable Functions

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