# Questions tagged [hilbert-spaces]

For questions involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.

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### $x_0+M\subset K$ for some linear subspace $M$ of a Hilbert space implies $x_0$ is orthogonal to $M$, where $K$ is a closed convex set

Suppose $K$ is a closed convex set in a Hilbert space $H$. If $x_0+M\subset K$ for some linear subspace $M$ of $H$, prove that $\langle x_0,y\rangle=0$ for all $y\in M$; in other words, $x_0$ is ...
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### Isomorphism between tensor product of Hilbert spaces

While I am familiar with the isomorphism between finite-dimensional vector spaces with the same dimension and isometric isomorphism between Hilbert spaces, I realize I am getting confused by $n$-th ...
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### Showing that an integral operator is bounded on a hilbert space.

Im trying to understand how it was hsown that T is a bounded operator on H. Going through the proof I don't understand what justifies the convergence of the last integral in this proof? What is it ...
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### Density problem involving embedded Hilbert spaces

It is well knon that if $s, t\in\mathbb{R}$, with $s >t$, the following continuous embedding holds $$H^s(\mathbb{R}^n)\hookrightarrow H^t(\mathbb{R}^n).$$ My question is: these spaces are also ...
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Let $U\in U(H)$ be unitary operator. Is it possible for it to has an empty point spectrum? I am aware that every bounded operator acting on complex Hilbert space has non-empty spectrum. Since $\... 0answers 28 views ### Prove that if$\sum\|e_n-f_n\|^2<\infty$, then$\{f_n\}$is complete. [duplicate] Let$\{e_n\}$and$\{f_n\}$be orthonormal sequences in a Hilbert space$H$with$\|e_n\|=\|f_n\|=1$and $$\sum_{n=1}^{\infty}\|e_n-f_n\|^2<\infty.$$ Prove or disprove: If$\{e_n\}$is ... 0answers 16 views ### Isonormal Gaussian process associated with a Hilbert space. We consider the isonormal Gaussian process$W=\{W(h),h\in H\}$indexed by a separable Hilbert space$H$, defined on a complete probability space$(\Omega, \mathcal F,P)$where$\mathcal F:=\sigma(W)$,... 1answer 13 views ### Eigenvectors of hermitian matrix form unitary matrix Let$A \in M_2(\mathbb{C})$be a hermitian matrix i.e.$A = A^*$. Suppose that$\lambda, \mu$are eigenvalues correspoing to normed eigenvectors$\begin{pmatrix} v_1 \\ v_2 \end{pmatrix}, \begin{...
Let $A$ be a self-adjoint (not necessarily bounded) operator on a Hilbert space $\mathscr{H}$ and $B$ be self-adjoint bounded. Then how would one show that $\sigma (A+B) \subseteq \sigma(A) +\sigma(B)$...