Let $V$ and $W$ be infinite-dimensional vector spaces over $\mathbb{C}$.
$(a)$ Is $X := \{f\in \operatorname{Hom}(V,W) :\, \operatorname{rank}(f) < \infty \}$ a subspace of $\operatorname{Hom}(V,W)$?
$(b)$ Is $Y := \{f \in \operatorname{Hom}(V,W) :\, \dim (\ker f) < \infty \}$ a subspace of $\operatorname{Hom}(V,W)$?
$(c)$ What is $X \cap Y$?
Attempt: $(a)$ Yes, $\operatorname{rank}(A+B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)$ gives closed under addition. Closure under multiplication is clear.
$(b)$ No, if $f: e_1 \mapsto 0, e_2 \mapsto e_2, e_3 \mapsto e_3,\dots$ and $g: e_1 \mapsto 0, e_2 \mapsto -e_2, e_3 \mapsto -e_3, \dots$ then $f+g$ has an infinite dimensional kernel.
$(c)$ $\dim (\ker f) + \operatorname{rank}(f) = \dim V$. Since $V$ is infinite dimensional, both $\dim (\ker f)$ and $\operatorname{rank}(f)$ cannot be finite. This means $X \cap Y$ is the empty set.
Is my solution correct?