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benf.stokes
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Homework Statement
Let H= C[-1,1] with L^2 norm and consider G={f belongs to H| f(1) = 0}. Show that G is a closed subspace of H.
Homework Equations
L^2 inner product: [tex]<f,g>\to \int_{-1}^{1}f(t)\overline{g(t)} dt[/tex]
The Attempt at a Solution
I've been trying to prove this for a while but i can't establish that given [tex]||f_{n}-f||< \epsilon[/tex] (where the norm is the L^2 one) we have uniform convergence for the sequence [tex](f_{n})[/tex]If I could prove this the result would follow easily given that G is contained in its closure and if [tex](f_{n})[/tex] converged uniformly we would have [tex]f(1)=\lim_{x \to 1} \lim_{n \to \infty} f_n(x)=\lim_{n \to \infty} \lim_{x \to 1} f_n=0[/tex] and thus that f belongs G
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