I'm confused about expressions like $L=\sum_{k=1}^{\infty} R_k, \quad \quad (1)$where $L:H\rightarrow H$ and $R_k:H \rightarrow H$ are continuous linear operators in a Hilbert space $H$. We assume $\sum_{k=1}^{\infty} R_k x$ is convergent every $x\in H$. The thing I'm confused about, is: If we know that for every $x\in H$ we have $Lx=\sum_{k=1}^{\infty} R_k x,$can we that automatically conclude that $(1)$ holds ?
I would conjecture "yes" since two maps are identical if they have the same domain and range (trivial here) and are identical for every argument - which they are, as the above shows.
But if the answer is indeed "yes", why is it in sum texts shown, that the $R_k$ additionally converge in the operator norm to $L$, i.e. $\left\|L- \sum_{k=1}^{n} R_k\right\| \rightarrow 0\quad (n\rightarrow \infty),$ before they conclude that $(1)$ is true ? Was that really necessary ?