# Tagged Questions

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

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### Corollary to Putnam's theorem

Suppose $T_1$ and $T_2$ are normal operators on Hilbert spaces $\mathcal H_1$ and $\mathcal H_2$, respectively. Putnam showed that if $X$ is an operator satisfying $T_2X=XT_1$, then $T_2^*X=XT_1^*$. ...
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### A problem with a proof of Bessel's inequality, and how to get Parseval's identity from it

I am studying functional analysis, and I think there is a problem with the proof written in my notes for Bessel's inequality. The theorem is: Let $H$ be a Hilbert space and ...
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### What is a good definition of Hilbert space?

Motivation of my question: in my opinion, in view of the common definition, the statement "$\ell_p$ is a Hilbert space if and only if $p=2$" makes no sense because there is no inner product in the ...
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### Is the fractional Laplacian on $\mathbb{R}$ of trace class or not?

Is the fractional Laplacian on $\mathbb{R}$ of trace class or not? I don't know the basis of $\mathrm{L}^{2}(\mathbb{R})$ for applying the definition of a trace class operator. Thank you very much.
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### real eigenvalues for non normal operator

Is there unbounded non normal operator in Hilbert space which has only real eigenvalues? If yes, could you give me an example?
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### l2 norms, rapidly decreasing functions and fourier transforms

Let $f\colon \mathbb{R} \to \mathbb{C}$ be a rapidly decreasing (rd) function. Let $\mathcal{F}(f)$ be the Fourier transform of $f$. It is known that 1) $\| \mathcal{F}(f) \|_2 = \| f \|_2$ ...
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### closedness a subset of a Hilbert space

Let $H$ be a Hilbert space that admits a countable orthonormal basis $\{e_i\}$. I know this means that $H$ is separable and so is $S$ (as a subset of it, defined below). Show that $S$ is a closed ...
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### Convergence of an operator in norm

Let $H$ be a Hilbert space and assume we have three converging sequences: $u_n\rightarrow u$ in $H$, $v_n\rightarrow v$ in $H$ and $\lambda_n\rightarrow \lambda$ in $\mathbb{C}$. I would like to prove ...
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### Showing that inner product of two vectors is the limit of the inner products

How can you show that $$(a,b) = (\sum_{i=1}^\infty a_i \phi_i,\sum_{i=1}^\infty b_i \phi_i) = \lim_{N\to \infty}(\sum_{i=1}^N a_i \phi_i,\sum_{i=1}^N b_i \phi_i)$$ where $\phi_i$ is an orthonormal ...
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### Equivalent norm in sobolev space H^2

I consider space $H^{2}(0,a)=\{ f\in L^{2}(0,a): f',f''\in L^{2}(0,a) \}$ I define norm $\Vert w \Vert_{H^{2}}:=b\Vert w''\Vert_{L^{2}}$, where b is positive constant. I couldn't proof that it is ...
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### Norms arising from all representations of *-algebras

It is common that in order to obtain a $C^*$-algebra from a $^*$-algebra $A$ one defines a norm on $A$ by $$\|x\|=\sup\{\|\pi(x)\|\,|\,\pi\ \text{is a }^*\text{-representation of }A\}.$$ However, I ...
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### How to determine a operator norm?

How to solve following: In Hilbert space $W_2^1=\{f:[0,1]\rightarrow \mathbb{C}|f\in AC[0,1], f'\in L^2[0,1]\}$ with scalar product $(f,g)=\int_0^1 f\overline{g}dx+\int_0^1 f'\overline{g'}dx$ is ...
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### Problems with understanding the proof for existence of projections to a close convex set on a Hilbert space

In the setting of an introduction to functional analysis course, I have read the following statement: Let $H$ be a Hilbert space and let $A\subseteq H$ be a closed convex set. Then there exist a ...
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### A Riesz representation theorem without coercivity

Let $b:H \times H \to \mathbb{R}$ be a bounded bilinear form on Hilbert space $H$. Fix $u \in H$. Then $b(u, \cdot):H \to \mathbb{R}$ is bounded so $b(u,\cdot) \in H^*.$ Then $b(u,\cdot) = F_u(\cdot)$ ...
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### An inner product on a space of linear maps

Let $V$ and $H$ be two complex Hilbert spaces. We suppose $V$ to be finite-dimensional. I'd like to understand the structure of Hilbert space on the space of linear mappings $\mathrm{Hom}(V,H)$. ...
### There exists a countable set of mutually orthogonal trigonometric functions which form a basis for $L^2(T)$. Proof?
Evidently, this fact (for real or complex valued functions) is usually taken "for-granted" in derivations of Fourier series/transform, taking $\{e^{inx}|n\in\mathbf Z\}$ as the set of basis vectors. ...