Prove that if $E \simeq E'$, $E \bigoplus F \simeq E' \bigoplus F'$ then $F \simeq F'$.

Question

All the spaces in this question are Banach Spaces, and "$\simeq$" means they are linear homeomorphic. The question is to prove the following:

Any hints on how to do this? I'm kind lost.

Thanks!

but the implication that $E \bigoplus F \simeq E \bigoplus F'$, how do i do it? I need to prove this! — Ulivai, Nov 25 '20 at 00:17
This is not true. Are you sure you have written the question correctly? — Eric Wofsey, Nov 25 '20 at 02:45
Yes, it's written correctly. the $\bigoplus$ is the direct sum of the two banach spaces. Why this isn't true? — Ulivai, Nov 25 '20 at 11:40

score 1 · Answer 1 · answered Jan 01 '21 at 04:28

1

A counterexample: $\ell_2\oplus\Bbb R$ is linearly homeomorphic to $\ell_2\oplus\Bbb R^2$, but $\Bbb R$ is not linearly homeomorphic to $\Bbb R^2$.

answered Jan 01 '21 at 04:28

Alex Ravsky

1 Answers1