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Results tagged with hilbert-spaces
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user 169474
A Hilbert space $H$ is a real or complex vector space endowed with an inner product such that $H$ is a complete metric space when endowed with the norm induced by this inner product.
2
votes
Wold decomposition of toral endomorphisms
The Hilbert space $H := L^2(\mathbb{T}^d,dx)$ can be canonically identified with the subspace of all (classes of) $\mathbb{Z}^d$-periodic locally square-integrable functions on $\mathbb{R}^d$. Then, f …
- 3,808
5
votes
Accepted
A nonlinear mapping on $L^2(S^1)$ that commutes with all translation operators is necessaril...
For $f \in H$, call $Mf$ the mean value of $f$ on $\mathbb{S}_1$. Let $\phi : \mathbb{R} \to \mathbb{R}$ be any non-Borel function. Call $\mathbb{1} \in H$ the constant function equal to $1$ everywher …
- 3,808