I'm interested in solving an equation of the form
$$ Ax = b $$
for some bounded linear operator $A: H_1 \mapsto H_2$ where $H_1, H_2$ are some Hilbert spaces.
I've seen in this math.SE post in particular, the claim that
the solution set should always be the kernel plus a particular solution.
Where does this claim come from?
I am also interested to learn in general about how operators in Hilbert spaces are introduced. Is any book on functional analysis good for this purpose?