In $\mathbb{R}^3$ using the dot product applied to two vectors $v,w$ we find $$ \langle v,w \rangle = \| v \| \| w\| \cos(\theta) $$
the same concept is in general separable Hilbert spaces. In $\mathbb{R}^3$ we can rotate a vector respect to a given axis using a rotation matrix $R_{\theta}$. I was wondering if there's a generalization of such geometric transformation that is expressed using the formalism of functional analysis (like function of operators, functional calculus etc).
Is there?
Update : here for example it is said:
For instance, the Fourier transform is a unitary transformation and can thus be viewed as a kind of generalised rotation.
Are then unitary transformations in Hilbert space a generalization of the rotations then? Are there any others geometric transformations that can be generalized?