Finite Rank Operator: Continuity

Question

I keep forgetting it, so...

Given Banach spaces $X$ and $Y$.

Then it is wrong: $$\dim\mathcal{R}F<\infty:\quad F\in\mathcal{L}(X,Y)\implies F\in\mathcal{C}(X,Y)$$

Can I construct such?

Answer 1

Ok satisfactory enough:

Given the Hilbert space $\ell^2(\mathbb{N})$.

Extend to a Hamel basis: $$\mathcal{B}:=\{\delta_n:n\in\mathbb{N}\}\cup\{\beta,\beta',\ldots\}$$

Define an operator by: $$T(\delta_n):=n\quad(n\in\mathbb{N})$$ $$T(\beta):=0\quad(\beta,\beta',\ldots)$$

Then it is unbounded.