I want to find a linear isometry $T:V\to V$ such that $T$ is not Bijective.
I think that, I need to considere a infinite dimensional space $V$, but I am not sure about these concepts.
Thanks for your help.
I want to find a linear isometry $T:V\to V$ such that $T$ is not Bijective.
I think that, I need to considere a infinite dimensional space $V$, but I am not sure about these concepts.
Thanks for your help.
Try $T:\ell^2 \to \ell^2$ where $T(x_1,x_2,\dots)=(0,x_1,x_2,\dots)$. It is injective (like all linear isometries), but not surjective.
Let $V$ be the real vector space whose basis is indexed by $\mathbb{N}$, and let $T:V\to V$ $v_i\mapsto v_{i+1}$. Then using the norm $||\sum_{i=1}^\infty a_iv_i||=\sum_{i=1}^\infty a_i^2$, $T$ is a linear isometry.