In need show that if $\mathcal{H}, \mathcal{K}$ are Hilbert spaces and $T:\mathcal{H}\to \mathcal{K}$ is a isometric linear operator. i.e, $\|T(x)\|=\|x\|$ for all $x\in\mathcal{H}$ then $T$ is a isomorphism between Hilbert Spaces, i.e:
Note: The inner product is defined on a field $\mathbb{K}$.
- $T$ is injective.
- $T$ is surjective.
- $\langle x,y\rangle=\langle Tx,Ty\rangle$ for all $x,y\in \mathcal{H}$.
I showed 1 and 3 but I can't to show part 2.
In 1, $T$ is injective from $T$ is isometry.
In 3, I showed using the polarization identity that $\langle x,y\rangle=Re(\langle Tx,Ty\rangle)+Im(\langle Tx,Ty\rangle)i=\langle Tx,Ty\rangle$. In this part is neccesary that I do separately the real case and complex case?
I Hope you can help me with part 2.
Best Answer
Hint:
It's an isomorphism between the Hilbert space $\ \mathcal{H}\ $ and the image of $\ T\ $ but not necessarily between $\ \mathcal{H}\ $ and $\ \mathcal{K}\ $, if $\ \mathcal{K}\ $ is merely some codomain of $\ T\ $. What you need to show is that the image of $\ T\ $ is a Hilbert space, which I don't think you'll find too difficult.