- #1
tom_rylex
- 13
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Homework Statement
(I'm posting this because my proofs seem to be lousy. I want to see if I'm missing anything.)
Show that if [tex] f_n \in L^2(a,b) [/tex] and [tex] f_n \rightarrow f [/tex] in norm, then [tex] <f_n,g> \rightarrow <f,g> [/tex] for all [tex] g \in L^2(a,b) [/tex]
Homework Equations
[tex] L^2(a,b) [/tex] is the space of square-integrable functions,
[tex] {f_n} [/tex] is a finite sequence of piecewise continuous functions, and
< , > is the inner product
The Attempt at a Solution
I started with a linear combination of inner products and applied the Cauchy Schwarz inequality:
[tex] \vert <f_n - f,g> \vert \leq \Vert f_n - f \Vert \Vert g \Vert [/tex]
By the definition of norm convergence, I have
[tex] \Vert f_n - f \Vert \rightarrow 0 [/tex], which means that
[tex] \vert <f_n - f,g> \vert \rightarrow 0 [/tex]
Since this is absolutely convergent, that means that
[tex] <f_n,g> - <f,g> [/tex] is also convergent to 0. So therefore,
[tex] <f_n,g> -<f,g> \rightarrow 0[/tex]
[tex] <f_n,g> \rightarrow <f,g> [/tex]
Is that reasonable? Or am I missing something?