Are finite-dimensional normed spaces injective?

Asked May 20 '17 at 15:30

Active May 20 '17 at 15:30

Viewed 47 times

A Banach space $X$ is said to be injective if every linear continuous $T:Y\to X$, with $Y\subseteq Z$ Banach spaces, has a linear continuous extension $T^*:Z\to X$ with the same operator-norm.

My question is: is every finite-dimensional normed space $X$ injective?

All examples I know are infinite-dimensional and besides they're not so simple.

Any hint? Thank you.

asked May 20 '17 at 15:30

Tanius

1,205

1

The $1$-dimensional version is surely the Hahn-Banach theorem. Are there any subtleties in going from $1$-dimensional $X$ to $n$-dimensional $X$? – Angina Seng May 20 '17 at 15:33
@LordSharktheUnknown got it, thank you – Tanius May 20 '17 at 16:38

Are finite-dimensional normed spaces injective?

0 Answers0