# Tagged Questions

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### Spectral decomposition of compact operators

Let T be a compact operator on an infinite dimensional hilbert space H. I am proving the theorem which says that $Tx=\sum_{n=1}^{\infty}{\lambda}_{n}\langle x,x_{n}\rangle y_{n}$ where ($x_{n}$) is an ...
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### Weak limits and subsequences

Let $S:X \to X$ be a (nonlinear) map between a Hilbert space $X$. I want to show that $S$ is weakly continuous, so if $x_n \rightharpoonup x$, then $S(x_n) \rightharpoonup S(x)$. To do this, I have ...
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### Positive compact operator has unique square root.

Let H be a hilbert space and T be a compact positive operator so that by the spectral decomposition theorem, $T=\sum_{n=1}^{\infty}{\lambda}_{n}\langle x,e_{n}\rangle e_{n}$ where the $e_{n}$ are the ...
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### Functional analysis in Banach space and Hilbert space

I would like to ask some questions. In Hilbert space, the inner product & norm are defined, so the orthogonal property, Pythagoras theorem and the adjoint operator can be defined. => It's clear. ...
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### Hahn-Banach separation theorem for Hilbert spaces

What is the strongest form of the Hahn-Banach separation theorem for Hilbert spaces? Could you please provide a reference?
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### Reproducing kernel Hilbert sapce

I encountered the following claim (verbatim): Theorem Let $V$ be a subspace of $L^2(\mathbb{R})$ and $\{e_n\}$ be a orthonormal basis of $V$. The $V$ is a reproducing kernel Hilbert space with kernel ...
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### Unbounded extension of bounded operator

Is it possible to construct an unbounded extension of bounded densely defined operator? To be more concrete, let $\mathcal{H}$ be Hilbert space, $\mathcal{D}\subset\mathcal{H}$ - a dense subset, ...
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### On the completeness of inner product spaces.

Let $H$ be a Hilbert space, equipped with an inner product $(\cdot,\cdot)_1$ and norm $\|\cdot\|_1$ induced by it. Let $(\cdot,\cdot)_2$ be other inner product on $H$ and $\|\cdot\|_2$ the norm ...
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### Condition for vector to be in the domain of unbounded operator.

Let $P$ be unbounded self-adjoint operator on some Hilbert space $\mathcal{H}$. We assume that the limit $$\lim_{\epsilon \searrow 0} \|\exp(-\epsilon^2 P^2/2) P\psi\|$$ exists and is finite. Does ...
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### Ultraweak topology on Banach spaces

If $X$ and $Y$ are Banach spaces with $Y$ reflexive, then the space $\mathcal{B}(X,Y)$ of bounded operators from $X$ to $Y$ is the dual of the projective tensor product of $X$ and $Y^{*}$. As in the ...
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### Spectral Theorem for normal operators

I want to prove this in the infinite dimensional Hilbert space case. What is the easiest way to go about this (What do I need to know, what theorems do I need,etc). My aim is to show every normal ...
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### Supremum calculation

Calculate $\sup(\sum_{k=n+1}^{\infty}\frac{|x_{k}|^{2}}{4^{k} })$, where $x=(x_{1},x_{2},....)$ is a member of $l_{2}$ and the supremum is take over all $x$ with $||x||= 1$. My intuition says the ...
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### Weak derivative of one parameter group and the domain of its generator

Let $U(t)=\exp(i t A)$ be a one parameter group generated by self-adjoint (unbounded) operator A. It is well-known that if $$\lim_{t\rightarrow 0} \frac{U(t)\psi-\psi}{t}$$ exists then $\psi$ ...
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### Distance between Unilateral shift and invertible operators.

I want to prove that the distance between unilateral shift and normal operators is $1$. But I need to prove that $d(S,\operatorname{Inv}(L(H))= 1$, where $H$ is a Hilbert space. Does anyone have any ...
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### How to find adjoint operator?

Let $(X,\langle\cdot,\cdot\rangle)$ be a Hilbert Space over $K$ with orthonormal basis $(x_n)$, and let $(\lambda_n)\in K$ be a bounded sequence. The mapping $T:X\to X$ is defined by ...
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### Derive Fourier transform from what it should do?

I was wondering about the following: Imagine you want to figure out whether there is a transform that exchanges differentiation with multiplication and convoution with pairwise transformation for ...
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### The trace class operators are the dual of the compact operators

I know that the map from the trace class operators $L_1(H)$ to the dual of the compact operators $K'(H)$ given by $A \mapsto tr( \cdot A)$ is an isometric isomorphism. Linearity is obvious by the ...
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### Orthogonal Projections in $U\subset V$ two subspaces in a Hilbert space

Let $U,V\subset H$ be closed subspaces of a Hilbert space $H$, and let $P_U$ and $P_V$ the respective orthogonal projections. Show: $U\subset V \Longleftrightarrow P_U=P_UP_V=P_VP_U$ Trying to ...
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### Can somebody explain the well-defined mapping to me?

Let $(X,(\langle.,.\rangle)$ be a Hilbert space over $\mathbb{K}$ with an orthonormal basis $(x_n)_{x\in\mathbb{N}}$ and let $(\lambda_n){n\in\mathbb{N}}\subset \mathbb{K}$ be a bounded sequence.The ...
### Is $H^2\cap H_0^1$ equipped with the norm $\|f'\|_{L^2}$ complete?
Let $-\infty<a<b<+\infty$. Consider the norms $\|\cdot\|_{L^2}$, $\|\cdot\|_0$ and $\|\cdot\|_{H^1}$ defined on suitable spaces and given by ...