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Let H be a Hilbert space. Prove that if [tex] \left\{ x _{n} \right\} [/tex] is a sequence such that lim[tex]_{n\rightarrow\infty}\left\langle x_{n},y\right\rangle[/tex] exists for all [tex]y\in H[/tex], then there exists [tex]x\in H[/tex] such that lim[tex]_{n\rightarrow\infty} \left\langle x_{n},y\right\rangle = \left\langle x,y\right\rangle[/tex].

The attempt at a solution

I'm pretty sure the idea is to show that [tex] \left\{ x _{n} \right\} [/tex] is Cauchy and hence convergent (since H is complete). Then [tex] x_{n} \rightarrow x [/tex] for some [tex] x \in H [/tex]. And since [tex] y \rightarrow y [/tex] and also the inner product is continuous, we have [tex] \left\langle x_{n},y\right\rangle \rightarrow \left\langle x,y\right\rangle[/tex].

I'm not convinced my argument for [tex] \left\{ x _{n} \right\} [/tex] being Cauchy is sound though and would appreciate any feedback:

[tex] \left\langle x_{m}-x_{n},y\right\rangle = \left\langle x_{m},y\right\rangle - \left\langle x_{n},y\right\rangle \rightarrow 0; m,n\rightarrow\infty[/tex].

And because this is true for all [tex] y \in H [/tex] we must have [tex] x_{m}-x_{n} \rightarrow 0[/tex] (this is where I think things might fall apart) and therefore [tex] \left\|x_{m}-x_{n}\right\| \rightarrow 0; m,n\rightarrow\infty[/tex]. And so [tex] \left\{ x _{n} \right\} [/tex] is Cauchy.

Unrelated to the above problem, I'd also really like to know why ( [tex] x_{n} \rightarrow x [/tex] ) is not properly aligned in the text. (This is my first time using Latex.)

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# Homework Help: Converging Inner Product Sequence in Hilbert Space

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