Show that for a sequence $(x_n)$ in an inner product space the conditions $\lVert x_n \rVert \to \lVert x \rVert$ and $\langle x_n,x \rangle \to \langle x, x\rangle$ imply convergence $x_n \to x$.
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$\|x_n-x\|^{2}=\|x_n\|^{2}-2\Re \langle x_n, x \rangle +\|x\|^{2}\to \|x\|^{2}-2\|x\|^{2}+\|x\|^{2}=0$.
Kavi Rama Murthy
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