I'm trying to solve an exercise in Brezis' Functional Analysis, i.e.,
Let $E, F$ be Banach spaces and $T:E\to F$ a compact bounded linear operator. We denote by $R(T)$ and $N(T)$ the range and kernel of $T$. Assume that $R(T)$ is closed.
- Prove that $T$ is a finite-rank operator.
- Assume, in addition, that $\dim N(T)<\infty$. Prove that $\dim E<\infty$.
There are possibly subtle mistakes that I could not recognize in below attempt. Could you please have a check on it?
- Because $F$ is a Banach space and $R(T)$ closed in $F$, we get $R(T)$ is a Banach space. Then we apply open mapping theorem for $T:E \to R(T)$ and get there is $\delta>0$ such that $$ B_{R(T)} \subset \delta T(B_E), $$ where $B_{R(T)}$ and $B_E$ are the open unit balls of $R(T)$ and $E$ respectively. Then $$ \overline{B_{R(T)}} \subset \delta \overline{T(B_E)}. $$
Because $T$ is compact, $\overline{T(B_E)}$ and thus $\overline{B_{R(T)}}$ are compact. Clearly, $\overline{B_{R(T)}}$ is the closed unit ball of $R(T)$. Then $R(T)$ is finite dimensional.
- By rank-nullity theorem, we get $\dim E = \dim N(T) + \dim R(T)$. The claim then follows.
