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

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2
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1answer
63 views

Show uniqueness in Hilbert-space.

In this theorem, I need to prove that $z_0$ is unique: Let $K$ be a proper closed linear subspace of a Hilbert space, and $x\in K^c$. Then there exists a unique $z_0 \in K^{\perp}$ such that ...
1
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1answer
43 views

Does the sequence $(\sqrt{n} \cdot 1_{[0, 1/n]})_n$ converge weakly in $L^2$?

Let $f_n (x) = \sqrt n 1_{[0,1/n ] } (x) \in L^2 (\mathbb R)$. Does $f_n$ converges weakly to $0$ in $L^2$? I tried to prove it by using Hölder's inequality or the Lebesgue differentiation theorem, ...
0
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2answers
36 views

How properties of a 2D hermitian matrix restrict the 2D matrix's elements

I have read different definitions, or properties, of a Hermitian matrix, and still am not sure if I have a sufficient number of properties to define a Hermitian matrix. Suppose the following is true ...
4
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1answer
71 views

Find a function $f$ such that $\int_{-\pi}^\pi xf(x) dx = \int_{-\pi}^\pi g(x)f(x)dx$

Problem Statement: We assume the following: $L^2[-\pi,\pi]$ is a real Hilbert space with the inner product $$\langle f,g\rangle = \int_{-\pi}^\pi f(x)g(x)dx$$ and the set ...
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0answers
36 views

Trace on non-separable Hilbert spaces

In a textbook I am reading the trace on operators of a Hilbert space is considered. Since in the book it is generally not assumed that the Hilbert spaces are separable, I wonder how the trace is even ...
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1answer
55 views

Coproducts of Hilbert spaces

Let $\mathsf{Hilb}_1$ (resp. $\mathsf{Hilb}_{\leq 1}$) denote the category of Hilbert spaces with linear isometries (resp. short linear maps). Does $\mathsf{Hilb}_1$ (resp. $\mathsf{Hilb}_{\leq 1}$) ...
4
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1answer
135 views

Hilbert space, dense, orthogonal complement.

Suppose $H$ is a Hilbert space, $A$ and $B$ are two subspaces. $A$ is closed and $B$ is dense. If $A^\perp \cap B=\{0\}$, or in other words, $\forall b\in B$, the projection to $A$ is not $0$, can we ...
1
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1answer
97 views

Proof of Closest Point Theorem in Hilbert Space

The theorem: Let $C$ be a non-empty closed convex subset of a Hilbert space $X$, and let $x \in X$. Then there exists a unique $y_0 \in C$ such that $||x-y_0|| \le ||x - y||$ all $y \in C$. In other ...
2
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1answer
54 views

$H$ is Hilbert, countable basis. If $||x_n|| \to ||x||$, and $\langle x_n,y\rangle \to \langle x,y\rangle \forall y\in H$. Show $||x_n-x|| \to 0$

Problem Statement: Suppose $H$ is Hilbert, with a countable basis. If $||x_n|| \to ||x||$, and $\langle x_n,y\rangle \to \langle x,y\rangle$ for all $y\in H$. Show $||x_n-x|| \to 0$. My attempt: I'm ...
4
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0answers
110 views

Inner product space of measures

Let $(X,\Sigma)$ be measurable space and $\mu_1,\mu_2,\dots$ set of finite measures on $X$ such that $\mu_i \perp \mu_j$ for $i\neq j$. Now we can consider space of measures: $$ \mathcal{M} = \left\{ ...
0
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1answer
69 views

Homogeneous Sobolev space is a Hilbert space

I am reading a book and I have some questions about the proof. The book wants to show $H^s(\mathbb{R}^d)$ is a Hilbert space iff $s<\frac{d}{2}$. $H^s(\mathbb{R}^d)$ is the homogeneous Sobolev ...
0
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2answers
59 views

Proving that an operator $T$ on a Hilbert space is compact

Let $H$ be a Hilbert space, $T:H \to H$ be a bounded linear operator and $T^{*}$ be the Hilbert Adjoint operator of $T$. Show that $T$ is compact if and only if $T^{*}T$ is compact. My attempt: ...
2
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2answers
57 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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1answer
54 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 ...
0
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1answer
29 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 ...
1
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1answer
39 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) ...
1
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1answer
36 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
votes
1answer
71 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: ...
0
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0answers
14 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
60 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}} ...
1
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2answers
48 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 $n=m$, ...
0
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1answer
36 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
34 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} ...
0
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1answer
53 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 ...
0
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2answers
63 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
96 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 ...
0
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1answer
72 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 ...
4
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1answer
54 views

Hilbert vs. De Morgan

Problem Given a Hilbert space $\mathcal{H}$. Then it holds: ...
0
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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
22 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 $y$ ...
1
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1answer
54 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 $T^*=T$ ...
1
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1answer
29 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
votes
1answer
70 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
31 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
130 views

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

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 ...
0
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0answers
12 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" $f(x)$. ...
3
votes
1answer
53 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
33 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?
0
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1answer
27 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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votes
2answers
103 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
votes
1answer
50 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
43 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 ?
0
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1answer
21 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
31 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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1answer
43 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
votes
1answer
101 views

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

Given a function $f$, we can write $f\colon\mathbb{R} \to \mathbb{R}$ to denote that $f$ takes a number from $\mathbb{R}$ into $\mathbb{R}$. Easy enough. Given the Laplace transform operator ...
3
votes
1answer
66 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
votes
2answers
206 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
votes
0answers
39 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 ...
1
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1answer
45 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 ...