direct sum of image and kernel in a infinitedimensional space Is it true that in an infinitdimensional Hilbert space the formula $$\text{im} S\oplus \ker S =H$$holds, where $S:H\rightarrow H$ ?
I know it is true for finitely many dimensions but I'm not so sure about infinitely many. Would it be true under some additional assumption, like assuming that the rank of $S$ is finite ?

Consider $l_2$, and the translation operator $T:~e_n\mapsto e_{n+1}$, it's injective but not surjective. So that $ker~T=0,im~T\neq l_2,ker~T\oplus im~T\neq l_2$.
If $rank~im T$ is finite, i remember i have learned that the equality holds in some book. (but it's vague to me now, so take care)
You can try to consider the coordinate function on the finite dimensional space $im~T$ in this case.
Other kinds of additional assumptions may concern with 


*

*Projection operator

*Compact operator

*Fredholm operator
Try your best
