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

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21 views

Convergent Operator, weakly convergent sequence => weakly convergent?

Suppose we have a Hilbert space $X$, a weakly convergent sequence $u_k\rightharpoonup u$ and a convergent operator $T_k \rightarrow T$ in the norm of $\mathcal{L}(X)$ (bounded, linear operators). Is ...
3
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0answers
23 views

Proving that Eigenvectors Span Hilbert Space

I have a specific problem I am trying to solve, but I would like to learn general principles, so I will start my question pretty general and add specifics later. Please answer the most general form of ...
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1answer
24 views

A vector in a function space

Suppose we let $$L^{p=2}(D,\mathbb{R})$$ denote a set of real functions on a domain D such that if $$\mathbf{a} \in L^{p=2}(D,\mathbb{R})$$ then we have $$\int_{D} \left | a(t) \right ...
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0answers
29 views

Connection between Hilbert space and Lie group. [on hold]

A vector space consists of two groups, one for addition and one for multiplication. How are Hilbert space and Lie group connected? Are the representations of a Lie group in Hilbert space, or do we ...
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1answer
27 views

Choosing a smooth function with desirable properties

Consider a smooth function $\varphi \in C^\infty[0, 1]$, where $\varphi (1) = 0$. My question is, can we necessarily choose another function $\psi \in C^\infty[0, 1]$, such that $\psi \geq 0, \psi(1) ...
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1answer
20 views

Riesz representative of gradient of $f(u) = u^*u$ in different inner products

This is a seeming "paradox" that has been bothering me for some time, as it (or other situations like it) show up often when computing gradients for numerical optimization on complex vector spaces. ...
8
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1answer
44 views

A sequence that converges weakly but not in the Cesàro sense

Let $H$ be a Hilbert space over $\mathbb{C}$ with inner product $\langle\cdot,\cdot\rangle$, and let $\{x_n\}_{n=1}^\infty\subseteq H$, $x\in H$. I'm using the following definitions: ...
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0answers
8 views

Tensor product of infinite-dimensional Hilbert spaces and tensor product of $\mathbb{C}$-modules.

In here I found the following construction. Let $R$ be a commutative ring, and $M,N$ be $R$-modules. The set $M\times N$ is well defined, and it is the starting point of the definition of the tensor ...
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1answer
51 views

Why is $\overline{\operatorname{span}\{e_n\mid n\in\mathbb{Z}\}}=L^2(\mathbb{T})$?

I want to know, why $\{e_n\mid n\in\mathbb{Z}\}$ is an orthonormal basis of $L^2(\mathbb{T})$, where $\mathbb{T}=\{z\in\mathbb{C}\mid |z|=1\}$, $e_n(z)=z^n$, and $\int_{\mathbb{T}} ...
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2answers
42 views

Orthonormal basis of $L^2(T)$

Why is $\{e_n\mid n\in\mathbb{Z}\}$ an orthonormal basis of $L^2(T)$, where $T=\{z\in\mathbb{C}\mid |z|=1\}$, $e_n(z)=z^n$, and $\int_T f(z)\,dz:=\int_0^1f(e^{2\pi i t})\,dt$? My try: If $n=m$, ...
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1answer
32 views

Euclidean geometry and $L_2(\lambda)$ space

Suppose $f,g\in L_2(I,\lambda)$ with $\lambda$ any probability measure and the norm $\| x\|=\sqrt{\langle x, x\rangle}$. Could we have the same geometric properties in this space as in the Euclidean ...
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1answer
23 views

quadratic form in hilbert space and Gram matrix

We are in Hilbert space $L^2$ we are given a subspace of dimension K as $$ V=\{ g_k,1 \le k \le K \} $$ everything that folows is defined on $V$ we define map $$ x \mapsto Q(x):= \sum_{k=1}^{K} ...
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1answer
31 views

Quadratic form in Hilbert space associated with orthogonal projection operator

we are in Hilbert space $L^2 $ and we are given subspace of dimension $2K$ $$ V=Vect\{ g_k,\bar{g_k},1\le k\le K \}$$ $V$ is a sum of $K$ subspaces of dimension 2 $$ W_k=Vect \{g_k,\bar{g_k} \} $$ now ...
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2answers
36 views

Weak convergency vs strong convergency in Hilbert space

Let $\mathcal{H}$ be an Hilbert space and let $(x_n)_n \subset \mathcal{H}$ be a sequence s.t. $$ x_n \rightharpoonup x ~~~,~~~ \| x_n \| \to \|x\| $$ We want to show that $ x_n \to x $. Now, I ...
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2answers
50 views

Volterra-like operator is bounded

Define $T:L^2(\mathbb R) \rightarrow L^2(\mathbb R)$ by $$(Tf)(x)=\int_{-\infty}^x e^{-(x-y)} f(y) \, dy.$$ I would like to show that $T$ is bounded and that $$\lambda = \frac{1}{1+iw}$$ is in its ...
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1answer
37 views

A closed subspace of a separable Hilbert Space is Separable

Suppose $X$ is a Hilbert Space which is separable. Let $Y$ be a closed Subspace of $X$. I need to show that $Y$ is separable. Since $X$ is separable it has a countable dense subset say $M$. Taking ...
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0answers
24 views

Orthogonal Complement: Families

Problem Given a Hilbert space $\mathcal{H}$. Consider a family: $$A:\Lambda\to\mathcal{P}(\mathcal{H}):\lambda\mapsto A_\lambda$$ Remind that: $$A\subseteq\mathcal{H}:\quad ...
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0answers
17 views

Calculate the real and the imaginary part of $g_n=\phi_n-\sum_{i=0}^{n-1}\langle\phi_n,\phi_i\rangle g_i$

We have $\{\phi_n\}_{n=0}^\infty$ a linearly dense sequence of unit vectors in a Hilbert space $H$ (on $\mathbb C$). Define $$g_n=\phi_n-\sum_{i=0}^{n-1}\langle\phi_n,\phi_i\rangle g_i$$ Calculate ...
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1answer
21 views

To show that $y$ is the best approximation of $x$ from $G$ i.e $y$ is the unique element of $G$ such that $||x-y||=d(x,G)$

Let $G$ be a closed subspace of a Hilbert Space $H$. For $x \in H$, let $y$ be the orthogonal projection of $x$ on $G$. Then I need to show that $y$ is the best approximation of $x$ from $G$ i.e $y$ ...
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1answer
47 views

$(T_n)_{n\in\mathbb{N}}\subseteq L(H)$, $T_n\to T$ weak, why does there exist $C>0$ such that $\|T_n\|\le C$ for all $n\in\mathbb{N}$?

Let $H$ be a Hilbert space, $(T_n)_{n\in\mathbb{N}}\subseteq L(H)$ a sequence such that $T_n^*=T_n$ and $T_n\le T_{n+1}$ for all $n\in \mathbb{N}$. There exists a map $T\in L(H)$ such that $T^*=T$ ...
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1answer
16 views

Invariant subspace and projection

Let $F$ be a subspace of a Hilber space $H$, invariant under a bounded linear map $T$, and let $P$ be an orthogonal projection such that $Im(P)=F$. I need to show that $F$ and $F^\perp$ are ...
2
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1answer
34 views

No Hilbert space can have countable Hamel basis without using Baire's Category theorem

I have to prove that no Hilbert space can have countable Hamel basis just using the fact that any finite dimensional subspace is closed (more specifically without using Baire's theorem). I saw a paper ...
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1answer
26 views

Can someone help me to give some hints? Left Hilbert-$C_0(T,K(H))$ module $C_0(T,H)$

I tried to prove example 3.4 from the book Morita Equivalence and Continuous-Trace C$^*$-Algebras by Iain Raeburn and Dana P. Williams, but I get uneasy with notations and ideas. Let me restate my ...
8
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1answer
108 views

Sufficient Condition for $f\in L^{1}(\mathbb{R}^{d})$ to belong to $L^{2}(\mathbb{R}^{d})$

Question. Let $\left\{\varphi_{j}\right\}$ be a complete orthonormal system for $L^{2}(\mathbb{R}^{d})$ such that each $\varphi_{j}\in C_{b}(\mathbb{R}^{d})$ (the space of continuous, bounded ...
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0answers
9 views

Variational function versus variational solution

I want to minimize the functional $F[f(x)]$ and I'm going to try this in two different ways: First I am going to numerically minimize the functional $F[f(x)]$, leading to the "true solution" $f(x)$. ...
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23 views

A is an Hilbert operator, $(A+I)^{-1}$ is continuous with dense domain then A is essentially self-adjoint

We have no information about A, just that is an operator defined in $X\subset H$ where $H$ is a Hilbert space.
3
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1answer
50 views

$f\mapsto \frac{df}{dx} - \frac{x}{\sqrt{1+x^2}}f $ has closed image and $1$-dimensional cokernel

Let $X$ be the completion of the space of smooth, compactly supported real-valued functions on $\mathbb R$ under the norm $$\|f\|_X^2=\int_{\mathbb R} \left(\frac{df}{dx}\right)^2 + f^2.$$ Let ...
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2answers
24 views

Complete eigen-vector basis from non invertible linear application

Consider a non-invertible linear application $O$ acting on a Hilbert space (quantum mechanics). Is there still any chance to find a complete basis of $O$ eigen-vectors or no?
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1answer
19 views

Direct Integral: Measurability

Given a Borel space $\Omega$. Consider plain functions: $$\eta,\vartheta\in\mathcal{F}(\Omega):=\{\eta:\Omega\to\mathbb{C}\}$$ The implication is wrong: ...
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2answers
75 views

Does orthogonal decomposition characterize direct sums in Hilbert space?

Let $H$ be a Hilbert space with inner product $(\cdot, \cdot)$. I know that if $M$ is a closed subspace of $H$, then $H$ can be written as the direct sum $M \oplus M^\perp$, where $M^\perp$ stands ...
2
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1answer
49 views

Integral Measures: Identification

Problem Given a Borel space $\Omega$. Consider a Borel measure: $$\mu:\mathcal{B}(\Omega)\to\overline{\mathbb{R}}:\quad\mu\geq0$$ Regard a Borel measure: ...
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1answer
33 views

Dual of Hilbert space : induced norm vs. operator norm

Let $\mathfrak{H}$ be a Hilbert space. Is the operator norm on the dual $\mathfrak{H}^*$ induced by a inner product ?
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1answer
17 views

$\lim_{k \to \infty} \langle s_k,e_n \rangle = \langle h,e_n \rangle$

I took a passage from a textbook regarding equivalent conditions of having an orthonormal sequence in a Hilbert space H. Why is the equality $$\lim_{k \to \infty} \langle s_k,e_n \rangle = \langle ...
2
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1answer
28 views

Partial Isometries: Final

Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$J\in\mathcal{B}(\mathcal{H},\mathcal{K}):\quad P:=J^*J$$ By the C*-property: $$J=JJ^*J\iff P^2=P=P^*$$ Note that in any ...
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0answers
17 views

Simple inner product relation for self-adjoint operators

Problem 6.3.7 in Friedmann's Foundations of Modern Analysis asks to show that if $A$ is a self-adjoint operator in a Hilbert space then \begin{align} 4(Ax,y) = &\left[\left(A(x+y),x+y\right) - ...
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1answer
41 views

Need help proving $n(T)=n(T^*)$ for finite dimensions.

In my book this is showed: Let H and K be complex Hilbert spaces and let $T\in B(H,K)$. There exists a unique operator $T^* \in B(K,H)$ such that $(Tx,y)=(x,T^*y)$ for all $x\in H$ ...
2
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1answer
55 views

A very simple question: what spaces of function does the laplace transform map from and into?

Given a function $f$, we can write $f:\mathbb{R} \to \mathbb{R}$ to denote that $f$ takes a number from $\mathbb{R}$ into $\mathbb{R}$. Easy enough. Given the laplace transform operator ...
3
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1answer
59 views

Solution to Equation $Ax=f$ in Hilbert Space

Question. Let $H$ be a separable Hilbert space with complete orthonormal basis $\left\{u_{k}\right\}_{k=1}^{\infty}$, let $H_{n}:=\text{span}\left\{u_{1},\ldots,u_{n}\right\}$, and let ...
3
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2answers
192 views

Power series expression for $\exp(-\Delta)$

I know it should be true, but for some reason I can't get the calculations to work out in order to show that if $f$ is smooth and compactly supported, the power series $\sum_{j=0}^\infty ...
2
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0answers
28 views

Absolutely Continuous Spectrum and Norm of Resolvent

Problem. Let $H$ be a Hilbert space, and let $A:H\rightarrow H$ be a bounded, linear operator. Suppose $A$ has purely absolutely continuous spectrum and $\sigma_{ac}(A)=[0,1]$. Find the set of ...
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1answer
19 views

Getting the unique element in the Riesz-Frechet Theorem.

I have this thorem in my book, H', denotes the dual space, that is the set of bounded linear operators from X to the field over X. The way they got the unique element seems very interesting. Does ...
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3answers
76 views

Stone's theorem for bounded operators

Let $H$ be a Hilbert space (assume separable if you like), and let $(U_t)_{t\in\mathbb{R}}$ be a unitary representation of $\mathbb{R}$ on $H$. Let us assume that $t\mapsto U_t$ is continuous, where ...
2
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2answers
27 views

Adjoint of $\lambda I - T$

Given a selfadjoint (maybe unbounded) operator $T$ on a Hilbert space $H$, I want to calculate the adjoint of $\lambda I - T$ for a $\lambda \in \Bbb C$. I am tempted to argue as follows: ...
3
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1answer
73 views

Spectral resolution of multiplication operator

Kosaku YOSIDA claims in his book "Functional Analysis" that it is easy to see that the multiplication operator $Hx(t) = tx(t)$ in $L^2(-\infty,+\infty)$ admits the spectral resolution $H = ...
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0answers
29 views

Operator norm of $P[v]-P[w]$

Let $\mathcal H$ be a complex Hilbert space with inner product $\langle\mid\rangle$, (dirac notation) which is semi-linear (conjugate linear) in the first argument. Let $\mathcal P_1=\mathcal P_1 ...
3
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1answer
30 views

Compact diagonal operator

Suppose $A : H \to H$, where $H$ is a Hilbert space, is bounded. Also, $A$ is a diagonal operator with diagonal $\{a_n\}$. Show: If $A$ is compact, then $a_n \to 0$ as $n \to \infty$. Should I prove ...
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1answer
43 views

Is there an error in the solution for this exercise?

I have this exercise: H is a complex hilbert space. And T is a compact operator on H. Show that if H is not separable, then 0 is an eigenvalue of T. Hint: Use lemma 1, and theorem 2. The ...
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1answer
46 views

Show that T is compact if $T^*T$ is compact

Let $H$ be a Hilbert space and $T: H\to H$ be a bounded linear operator. $T^*$ is the Hilbert adjoint operator. Show that $T$ is compact if $T^*T$ is compact. I am stuck with this proof. Any help ...
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0answers
45 views

Over ZF does “every non-seperable Hilbert space has an orthonormal basis” imply “there exists a non-Lebesgue measurable set”?

I know from this question that it's an open problem whether or not the existence of a dense orthonomral basis for every real or complex Hilbert space $(\text{B}_\text{orth})$ implies the axiom of ...
3
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0answers
18 views

Verification of a theorem regarding Mercer Kernel.

if $\langle x,y\rangle$ is a Mercer kernel, then is $\langle c_1 x_1 + c_2 x_2,y\rangle$ a Mercer kernel where $c_1+c_2=1$? Ans: I give the following (dirty) line of proof. Please tell whether its OK ...