Existence of nontrivial bounded linear operator?

Question

Are there two normed spaces such that there is no nontrivial bounded linear operator between them: $$\nexists T:X\to Y: T\text{ nontrivial, linear and bounded}$$

Answer 1

The Hahn-Banach Theorem gives us a wealth of nontrivial bounded linear functionals on (nontrivial) normed linear space $X$. We can for example take the "coordinate" map on a one-dimensional subspace $\{ru | r \in \mathbb{R} \}$, $f(ru)=r$, and extend it to a map $F:X \to \mathbb{R}$ also of the same norm as $f$.

Then another nontrivial bounded map $G:\mathbb{R} \to Y$ can be defined by $G(r) = ry$ for any nonzero vector $y \in Y$. The composition $T = G \circ F$ is then a nontrivial bounded linear transformation from $X$ to $Y$.