My apologies in advance for asking what (to me) looks like an extremely stupid question, but I just can't figure it out.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data:

Where is this an inner product:

[tex]\int_{a}^{b}f(x)g(x) dx[/tex]

a) on C[a,b]?

b) on C(R)?

The answer is that it is an inner product on a), but not on b) - apparently on b) the axiom of positivity fails. I do not understand how this is possible, since all functions that are C(R) are also C[a,b] - or have I just always misunderstood this notation? Does not C(R) mean "functions continuous on all of R"?

2. Relevant equations

This is what the answer key says: It fails on b) because [tex]\exists f \ne 0 : \Vert f \Vert^2 = 0[/tex]

3. The attempt at a solution:

has mainly consisted of trying (unsuccessfully) to work backwards towards a counterexample. I don't know how to do this in any other, more general way, since the whole idea seems illogical to me. Please enlighten me, somebody?

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# Homework Help: Inner product on C(R)

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