Let $L:H\rightarrow H$ be a continuous linear operator and $R_n:H \rightarrow H$ a sequence of continuous linear operators, where $H$ is a Hilbert space. If the $\sum_{n=1}^{k} R_n$ converge pointwise to $L$, meaning for every $x\in H$ we have $Lx=\sum_{k=1}^{\infty} R_k x=\lim_{k\rightarrow \infty}\sum_{n=1}^{k} R_nx,$ is then do we know then, that the operator $T$ defined by $ Tx=\sum_{k=1}^{\infty} R_kx,$
1) equals L ? 2) is continuous ? 3) convergences in the operator norm to $L$ ?
( For 1) and 2) with the help of Asaf Karagila, I think the answer is "yes", since for 1) $L$ and $T$ agree for every value and for 2) we know that $L$ was continuous, so using 1), also $T$ has to be continuous, right? )
EDIT: Everything has been cleared except that I need a counterexample for 3)