TITLE: Cauchy+pointwise convergence $\Rightarrow$ uniform converges (for an operator in a Hilbert space)
QUESTION [5 upvotes]: Suppose that the sequence of operators in a Hilbert space $H$, $\left(T_{n}\right)_{n}$,
is Cauchy (with respect to the operator norm) and that there is an
operator $L$, such that $Lx=\lim_{n\rightarrow\infty}T_{n}x$, for
all $x\in H$ (i.e. the $T_{n}$ converge pointwise to $L$).
How
can I prove then, that $\left(T_{n}\right)_{n}$ converges with respect
to the operator norm to $L$ (i.e. $\left(T_{n}\right)_{n}$ converges
uniformly to $L$)?

REPLY [2 votes]: The space of bounded linear operators under the operator norm is complete.  Hence, since $\{T_n\}$ is Cauchy, there exists an operator $T$ such that $T_n \to T$ in operator norm.  In particular, $T_n \to T$ pointwise, since $\|T_n x - T x\| \le \|T_n -T\| \|x\|$.  But $T_n \to L$ pointwise also, so we must have $T=L$.