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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?

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    $\begingroup$ Usually, someone who is starting functional analysis is well aware of the idea of a unitary/orthogonal transformation, at least in the finite dimensional context (over an inner product space). Are these ideas new to you? $\endgroup$
    – Ben Grossmann
    Aug 14, 2017 at 14:34
  • $\begingroup$ Probably the most important "types of transformations" that generalize from $\Bbb R^3$ to the infinite dimensional context are rotations/reflections, orthogonal projections, and positive (definite) transformations. $\endgroup$
    – Ben Grossmann
    Aug 14, 2017 at 14:36
  • $\begingroup$ @Omnomnomnom I know linear algebra, so those ideas are not new to me. However of course I struggle a bit to understand what's the equivalent in functional analysis. To me linear algebra deals with matrices, while functional analysis deals with integral/differential operator, therefore it's a bit hard to imagine a "Rotation integral" o a "Rotation differential operator" without actually understanding what happens. $\endgroup$
    – user8469759
    Aug 14, 2017 at 14:40
  • $\begingroup$ But surely you've seen an orthogonal/unitary matrix which wasn't $3 \times 3$. I'd say that, coming from $\Bbb R^3$, imagining a "rotation" in $\Bbb R^{57}$ is just as hard as imagining a "rotation integral". $\endgroup$
    – Ben Grossmann
    Aug 14, 2017 at 14:43
  • $\begingroup$ I didn't mean to be this literal... I'm just curious to know how these generalizations actually look like in formulas and whether or not these are bounded/unbounded, spectral decomposition eventually etc. $\endgroup$
    – user8469759
    Aug 14, 2017 at 14:44

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It's hard to tell exactly what you're looking for, but here's an "example" that might appeal to you. Let $u,v$ be an arbitrary orthonormal pair of vectors in a Hilbert space $H$. We can define a rotation by angle $\theta$ in the $uv$-plane by defining $R_\theta : H \to H$ by $$ R_\theta(x) = x + (\cos \theta - 1)[\langle x,u \rangle u + \langle x,v \rangle v] + \sin \theta[\langle x,u \rangle v - \langle x,v\rangle u]\\ = x + [(\cos \theta-1)\,u + \sin \theta\,v ]\langle x,u \rangle + [(\cos \theta-1)\,v - \sin \theta\,u ]\langle x,v \rangle $$ For instance, if we have $H = L^2([0,1])$, we might write $$ R_{\theta}(f) = f(t) + [(\cos \theta-1)\,u(t) + \sin \theta\,v(t)]\int_0^1 f(\tau)u(\tau) \,d\tau\\ + [(\cos \theta-1)\,v(t) - \sin \theta\,u(t)] \int_0^1 f(\tau)v(\tau) \,d\tau $$

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