$B$ be an uncountable dimensional real Banach space and $T:B \to B$ be a surjective isometry such that $T(0)=0$ , then is $T$ linear?

Question

Let $B$ be a real Banach space and $T:B \to B$ be a surjective isometry such that $T(0)=0$ , then is $T$ linear ?

I know that it is true if $B$ is a Hilbert space ( we only need an inner product actually and even without assuming $T$ is surjective , [ though if $B$ is a finite dimensional NLS then any linear isometry on $B$ is surjective] ) , but have no idea what happens for Banach spaces in general . Please help . Thanks in advance

Answer 1

A more general case is when $B$ is a real normed space. You vcan apply Mazur–Ulam theorem :

If ${\displaystyle V}$ and ${\displaystyle W}$ are normed spaces over $\mathbb{R}$ and the mapping ${\displaystyle f\colon V\to W} $ is a surjective isometry, then ${\displaystyle f}$ is affine.

So $f$ is affine with $f(0)=0$, so $f$ is linear.

So if $B$ is a real Banach space, it is also true that $T$ is linear.

You can find a proof of the theorem on this paper : Proof of Mazur–Ulam theorem .