I was wondering what are the differences for, e.g. some vector space $V$ (over a field $\mathbb{k}$), between $\mathrm{End}(V)$, $L(V)$ and $B(V)$. I know this is a naive question but I would like to know your thoughts on this to improve my article writing.
I know that $L(V)$ stands for the $\mathbb{k}$-linear maps from $V$ to $V$, but also $\mathrm{End}_{\mathbb{k}}(V)$. However, the latter is more usual in e.g. representation theory or algebraic topology, but sometimes it has other meanings (for example, between algebra modules the endomorphisms have to respect that structure). On the other hand, many people use $B(V)$ when $V$ is finite-dimensional, which is sort of weird to me since $L(V) = B(V)$, although I can understand that this is because many statements only generalize for bounded operators. For example, for a finite-dimensional Hilbert space $\mathcal{H}$ would you employ $\mathrm{End}(\mathcal{H})$?
Thanks in advance!