Limits in inner product spaces

Question

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!

Answer 1

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 $n \to \infty$.