Asking for a counterexample of a bounded linear map which is not open

Question

I am self studying functional analysis and can someone please give an example of a bounded linear map which is not open .

Answer 1

Let $X$ and $Y\neq \{ 0 \}$ be a normed space. The map $L: X \rightarrow Y, Lx=0$ is linear, bounded and not open. Linearity and boundedness is clear and it is not open as $L$ maps the open set $X$ to $\{0\}$ which is not open (as $Y\neq \{0\}$).

Answer 2

The Open Mapping Theorem says if $X$ and $Y$ are Banach spaces, any surjective bounded linear map from $X$ to $Y$ is open.

Any linear map between normed linear spaces that is not surjective is not open.

If you want your counterexample to be surjective, let $X$ and $Z$ be Banach spaces, $T: X \to Z$ a bounded linear map whose range is not closed, and take $Y = \text{Ran}( T)$ with the norm of $Z$.