One of my homework questions asked me to show that $$(f, g) \mapsto \int_{0}^{1} f(x)g(x)dx$$ defines an inner product on $C([0,1])$, which I was able to do successfully. However, I am curious on whether or not $C([0,1])$ is complete if $\int_{0}^{1} f(x)g(x)dx$ is the inner product? I couldn't think of a counterexample, so I assume it is, but what might be a method to show this?
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asked Dec 6, 2022 at 0:27
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1$\begingroup$ The completion is $L^2$ $\endgroup$– ConradCommented Dec 6, 2022 at 0:45
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It is not. The norm would be $(\int_0^1 |f|^2)^{1/2}$ and it is easily to find a sequence of continuous functions for which the integral goes to $0$ yet the sequence is not convergent to (continuous) function which is $0$ on $[0,1]$.
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$\begingroup$ I don't see how it is easy to find such a sequence. Sorry if I am missing something obvious. $\endgroup$ Commented Dec 6, 2022 at 0:54
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$\begingroup$ Okay, I understand. Thanks for the clarifications. These sort of problems just aren't that intuitive. $\endgroup$ Commented Dec 6, 2022 at 5:50