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

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    $\begingroup$ Take a one-dimensional subspace of $Y$ and use a non-trivial bounded linear functional on $X$ to construct a non-trivial bounded operator from $X$ to this subspace. $\endgroup$ – David Mitra May 31 '14 at 15:10
  • $\begingroup$ Only for $X = \{0\}$ or $Y = \{0\}$. $\endgroup$ – Daniel Fischer May 31 '14 at 15:11
  • $\begingroup$ So there always exist nontrivial operators? $\endgroup$ – C-Star-W-Star May 31 '14 at 15:14
  • $\begingroup$ Can you explain the construction? $\endgroup$ – C-Star-W-Star May 31 '14 at 15:15
  • $\begingroup$ If you allow your norm to take value $\infty$, you could endow $Y$ with the (basically useless) norm $$ \| y\| = \begin{cases} 0 & y=0_Y \\ \infty & y \neq 0_Y. \end{cases}$$ In this case, the only bounded linear operator is trivial as long as there is some $0_X \neq x \in X$ such that $\|x\|< \infty$. $\endgroup$ – Tom May 31 '14 at 15:18

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 $Y$.

  • $\begingroup$ As answered in the Comments by David Mitra and Daniel Fischer. $\endgroup$ – hardmath May 31 '14 at 15:40
  • $\begingroup$ This doesn't show that any normed spaces admit nontrivial bounded linear operators! $\endgroup$ – C-Star-W-Star May 31 '14 at 15:49
  • $\begingroup$ Why doesn't it? $\endgroup$ – hardmath May 31 '14 at 15:53
  • $\begingroup$ Oh I'm sorry I overread the last sentence... $\endgroup$ – C-Star-W-Star May 31 '14 at 16:31

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