Convergence of a sequence of unit vectors in a Hilbert space

Question

Let $E$ be a vector space (over $\mathbb{R}$) with a positive definite hermitian form and let $\{x_{n}\}$ ($x_{n} \not= 0$) be a sequence converging to $x$ in the $L^{2}$ norm $\lvert\cdot\rvert=\sqrt{\langle\cdot,\cdot\rangle}$. Why does $\left\{x_{n}/\lvert x_{n}\rvert\right\}$ converge to $x/\lvert x \rvert$?

Answer 1

What you are really asking is why $|x_n| \rightarrow |x|$ the rest is just the limit theorem of elementary analysis.

Now we can use the inequality, which is an application of the triangle inequality,

$$||x_n|-|x||\leq |x_n-x|$$ to prove the limit.