# Convergence of a sequence of unit vectors in a Hilbert space

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 $x/\lvert x \rvert$?

• Short answer: The norm is continuous, scalar multiplication is continuous. Caveat: You must assume that $x\neq 0$, or things don't make sense. – Daniel Fischer Jun 13 '14 at 20:24

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