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Let $H$ be an inner product space.

I am stuck in a specific part in my exercise where I am supposed to show the following:

If the inner product $(x_n,y)$ converges to $(x,y)$ for all $y$ in $H$ and $\Vert x_n\Vert$ converges to $\Vert x\Vert$ then $x_n$ converges to $x$ as well.

I am very thankful for help!

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1 Answer 1

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By bilinearity of the inner product $$\left\| x - x_n \right\|^2 = (x-x_n,x-x_n) = \left\| x \right\|^2 + \left\| x_n \right\|^2 - 2 (x_n ,x) \to 2\left\| x \right\|^2 - 2 \left\| x \right\|^2 = 0,$$ as $n \to \infty$.

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    $\begingroup$ I'm assuming that the inner product is real-valued by the way. If it's complex valued, this should work with minor rephrasing. $\endgroup$ Commented Oct 16, 2016 at 18:43
  • $\begingroup$ Thank you! Could you also help me with the following statement: xn converges to x if and only if (xn,y) converges uniformly to (x,y) on the unit sphere around y. $\endgroup$
    – Yuhe
    Commented Oct 17, 2016 at 9:31

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