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fluidistic
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Homework Statement
Consider the space of continuous functions in [0,1] (that is C([0,1]) over the complex numbers with the following scalar product: ##\langle f , g \rangle = \int _0 ^1 \overline{f(x)}g(x)dx##.
Show that this space is not complete and therefore is not a Hilbert space.
Hint:Find a Cauchy sequence (with respect to the norm ##||f||=\sqrt {\langle f,f \rangle }## of functions in C([0,1])) such that the sequence converges to a non continuous function.
Homework Equations
Hmm.
The Attempt at a Solution
So I have in mind a sequence that reprent a truncated Fourier series that would represent a square wave if the series is never truncated. A kind of Heaviside function.
Or a sequence of Gaussians that would reprent the Dirac delta (but it's not a function).
In that case ##f_n=Ce^{-ax^2n^2}## would make it? I'm somehow confused when n tends to infinity to what happens at x=0.