I think Fourier approximation on step function is one example of incompleteness, is it true? Or could you suggest any intuitive examples for incompleteness?
2026-04-26 14:16:23.1777212983
Is continuous function space with standard inner product on $\big[0,\frac{1}{2}\big]$ not complete?
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It's not complete. Take a function $f(x)=1$ for $x \neq 0$ and $f(0)=0$. You can show that $f(x)$ can be well approximated by elements in $C[0,1]$ but $f$ is not in $C[0,1]$. The closure of $C[0,1]$ is actually what's called $L^2[0,1]$.