An example of unbounded isometry

Question

Suppose $H, K$ are two Hilbert spaces, $V: H\to K$ is an isometry if $V^*V=1$. Does there exist a concrete eexample of a unbounded isometry?

Answer 1

There does not exist such an example.

An isometry between Hilbert spaces is in particular an isometry as metric spaces. i.e. $\Vert V \Vert = 1$. Hence every isometry between Hilbert spaces is bounded.

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An isometry preserves the norm since $\forall x \in H$

$$ \Vert A x \Vert^2 = \langle Ax, Ax \rangle = \langle A^*Ax , x \rangle = \langle x,x \rangle = \Vert x \Vert^2 $$