Questions tagged [hilbert-spaces]
For questions involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.
7,663
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Interpreting the Definition of Gaussian White Noise
My book defines Gaussian white noise as an isometry from $L^2(X,\mathscr{F}, \mu)$ to a vector space of Gaussian random variables with mean zero. Can someone explain what on earth this isometry has ...
- 53
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1
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On the convergence of approximate units for C*-algebras.
Let $A$ be a non-unital C*-algebra and let $\pi : A \to \mathcal{B}(H)$ be a non-degenerate representation of $A$ (that is, $\mathrm{ span }\{\pi(a)h : a \in A, h \in H\}$ is a dense subset of $H$). ...
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What are the $\mathcal{M}_{sym}=\{E \operatorname{Lebesgue measurable and } E=-E \}$ measurable functions?
I am supposed to consider $L^2(-1,1)$ and the subspace $V= \{ u \in H : u \operatorname{is} \mathcal{M}_{sym}-measurable \}$ where $\mathcal{M}_{sym}$ is the $\sigma-$algebra generated by: $\{E \...
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Question on showing that $\frac{1}{n}\sum_{k=0}^{n-1}U^kf$ converges in norm to the orthogonal projection $Pf$ to the space $\{f\in H: Uf = f\}$
Edit: The reference I am reading is Yves Coudène's Ergodic Theory and Dynamical Systems, chapter 1, proof of theorem 1.1 on page 5.
Let $H$ be a Hilbert space and $U:H\to H$ a bounded linear operator ...
- 423
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1
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Integration over a finite-dimensional subspace of Hilbert space
Let $H$ be a separable Hilbert space with inner product $\langle,\rangle$, let $\{e_k\}_{k=1}^\infty$ be an orthonormal basis of $H$, and let $A: H\to H$ be a symmetric, positive definite and ...
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32
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Infinite decomposition of a Hilbert space
Let $H$ be a Hilbert space. Suppose we have $H_{n}\oplus G_{n}=H$ for all $%
n\geq 0$ so that $\left( H_{n}\right) _{n\geq 0}$ is an increasing sequence
of closed subspaces and $\left( G_{n}\right) _{...
- 1,001
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An orthonormal basis of $L^{2}(\mathbb{R}^{n})$ which is pointwise in $\ell^{2}(\mathbb{N})$?
Does there exist an orthonormal basis of the Hilbert space $L^{2}(\mathbb{R}^{d})$, say $(e_{n})_{n=1}^{\infty}$, such that all elements $e_n\in L^{2}(\mathbb{R}^{d})\cap C^{0}(\mathbb{R}^{d})$ and ...
- 207
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Is there analog of angle in complex domain?
Given a real vector space with an inner product, i. e. a positive definite symmetric bilinear form $\beta$, I can define angle between any two nonzero vectors $v_1$, $v_2$ as $\arccos\frac{\beta(v_1,...
- 1,533
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Elias Stein's Real analysis chapter 4 exercise 9
Elias Stein's book "Real Analysis, Measure Theory, Integration, Hilbert spaces" chapter4 Exercise 9:\
let $H_1 = L^2([-\pi,\pi])$ be the Hilbert space of functions $F(e^{i\theta})$ on the ...
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3
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Non-unitary isometry and a norm equality
I am looking at a paper which asserts the following equality relating a non-unitary isometry. There is no explanation given for this, and I cannot figure out why this is true:
Here is the proposition: ...
- 175
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1
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Show that $K(u)=\sum_{k=0}^r P_k(0) P_k(u) \mathbf{1}_{\{|u| \leq 1\}}$ is a kernel of order $r$
I have a question concerning the construction of kernels wit orthogonal polynomials. The instructor defined the orthogonal polynomials as
$$P_0(x)=\frac{1}{\sqrt{2}}, P_m(x)=\sqrt{\frac{2 m+1}{2}} \...
- 165
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Closed Hilbert half-space
Suppose $H$ is the seperable infinite-dimensional real Hilbert space and $f$ is a continuous linear functional on it. Is the closed half-space $H_{f \ge 0} = \{ x \in H | f(x) \ge 0 \}$ homeomorphic ...
- 1,814
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Definition of the closure of an operator
I am reading Reed & Simon - Methods of Modern Mathematical Physics book and, in Chapter VII, the notion of an unbounded linear operator is introduced. Let $\mathscr{H}$ be a Hilbert space and $T$ ...
- 1,601
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Hilbert space decomposition through tensor product
I'm trying to understand how to decompose a high $N$-dimensional Hilbert space $V$ thinking it as a subspace of the tensor product space $V_1 \otimes V_2$ of two smaller Hilbert spaces $V_1$ and $V_2$,...
- 21
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Independence in functional theoretic view on probability
I'm reading the book Hilbert Space Methods in Probability and Statistical Inference by Small and McLeish.
There (Def. 3.2.4) independence is defines as follows
Let $\mathbf{B}$ and $\mathbf{C}$ be ...
- 429
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$\overline{\operatorname{Span}(A)} = \{ \sum \lambda_n a_n \}$?
Let $H$ be a separable Hilbert space on $\mathbb K = \mathbb R$ or $\mathbb C$. Consider $A \subset H$ a subset (not necessarily a linear one), do we have
$$ \overline{\operatorname{Span}(A)} = \left\...
- 1,840
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Definition of a stochastic process with orthogonal increments
Let $(\Omega,\mathcal{A},P)$ be a probability space and $X_t:\Omega \to \mathbb{R}$ be a stochastic process indexed by, e.g. $t \in [0,\infty)$.
Then $X = (X_t)_{t \geq 0}$ is said to have orthogonal ...
- 2,449
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$\{x \in H: \overline{U(H)x}~ \text{ is compact}\}$ is a closed subset of $H$, where $U(H)$ denotes the space of unitary operators on $B(H)$
Let $H$ be a Hilbert space and $U(H)$ denotes the space of unitary operators on $B(H)$. For $x \in H$, we define $U(H)x :=\{Ux: U \in U(H)\}$. Now consider the subset $\Omega \subseteq H$ as $\Omega :=...
- 1,775
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How to obtain the feature map from this kernel function
Here in this paper:
https://ideas.repec.org/a/eee/ejores/v292y2021i3p1004-1018.html ,
the author has written that $\phi (\cdot )$ represents the feature map from input space $\mathcal{X}$ to the high ...
- 2,575
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Nets and closedness of $\mathcal{B}(H)$ in the topology of pointwise convergence
Let $H$ be an infinite-dimensional Hilbert space and $\mathcal{B}(H)$ the set of bounded linear operators on $H$.
One way to define the strong operator topology (SOT) on $\mathcal{B}(H)$ is by ...
- 4,530
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1
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Is $A\in B\left( \overline{H}\right) $ invertible if it is onto and $\left. A\right\vert _{H}\in B\left( H\right) $ is one-to-one and onto?
Let $\overline{H}$ be a Hilbert space and $A\in B\left( \overline{H}\right) $
be a surjective operator such that $A\left( H\right) =H$ and its restriction
to $H$ is injective and. I'm asking if under ...
- 1,001
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1
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Why $H^1(0,1) = L^2(0,1)$ is not good?
From the theory of Hilbert bases and separability considerations one can easily derive a Hilbert space isomorphism between $H^1(0,1)$ and $L^2(0,1)$. I am convinced that such an isomorphism is useless:...
- 1,840
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Inner products and unbounded linear operators in Hilbert spaces
Suppose I have four compact, injective linear operators on a Hilbert space $A$, $B$, $C$, and $D$ and two elements of the Hilbert space $a$ and $b$. We know that $AB^{\star}=CD^{\star}$, $C[a]$ is in ...
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$L^{\infty}$ norm of a gradient vector
I am reading this from a paper and I do not understand the meaning of the notation. Hope that someone could explain it to me.
Let $a(x,y):D \times U\to \mathbb{R}$, where $D, U\subset\mathbb{R}^d$. $a(...
- 141
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Papa Rudin $4.22$ Theorem
There is the theorem:
Every orthonormal set $B$ in a Hilbert space $H$ is contained in a maximal orthonormal set in $H$.
There is the proof:
Let $\mathscr P$ be the class of all orthonormal sets in $H$...
- 801
2
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answers
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Compact operators evaluated on the sequence of eigenvalues, part II
As suggested in a comment below my previous question here I'm going to ask a similar question but under a more specific hypothesis.
Let $\Omega\subset\mathbb{R}^n$, $n\geq 2$ be a bounded domain with ...
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Generalization of the lemma, which connects absolute convergence and unconditional convergence of the series in Hilbert Space
I consider equivalence of absolute convergence and unconditional convergence in Hilbert space for $X = \mathbb{C} $
Lemma: If $(c_{n})$ is a sequence of real or complex scalars, then
$\sum_n c_n$ ...
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$\underset{n\rightarrow\infty }{\lim}\Vert \underset{i\geq 0}{\sum }\lambda _{i}^{n}x_{i}\Vert^{\frac{1}{n}}\not=0$ if $\lambda _{i}\not=0$?
Let $H$ be a Hilbert space. Suppose we have a basis $\left( x_{i}\right)
_{i\geq 0}$ for $H$ (not necessarily orthonormal) and a sequence $\left(
\lambda _{i}\right) _{i\geq 0}$ in $\mathbb{C}$, such ...
- 1,001
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The spectrum of the right shift operator on $\ell^2(\mathbb{Z}$)
The right shift operator $S$ on $\ell^2(\mathbb{Z})$ is defined such that for $x=\left(x_n\right)_{n \in \mathbb{Z}} \in \ell^2(\mathbb{Z})$, we have $S(x)_k=x_{k-1}$ for all $k \in \mathbb{Z}$.
...
- 1,485
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2
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If $\sup\| h_n \| < \infty$ and $\langle h_n, e\rangle \to 0$ then $\langle h_n, h \rangle \to 0$ for all $h$
Suppose $\mathscr H$ is a Hilbert space with orthonormal basis $\mathscr E$, and $\{ h_n \}$ is a sequence such that $\sup_{ n \in \mathbb N} \{ \| h_n \| \} < \infty$ and $\langle h_n, e\rangle \...
- 493
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2
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Hilbert Schmidt embedding (needed for Cylindrical Wiener processes)
In Karczewska's paper (2005) it says: "Assume that $U_1$ is an arbitrary Hilbert space such that $U$ is continuously embedded into $U_1$ and the embedding of $U_0$ into $U_1$ is a Hilbert-Schmidt ...
- 579
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If ${x_n}$ is weakly convergent to $x$ in $L^p(0,1)$ space, and if $\lim_{n}\|x_n\|=\|x\|$, then if $\lim_{n}x_n=x$? [duplicate]
If ${x_n}$ is weakly convergent to $x$ in $L^p(0,1)$ space, and if $\lim_{n}\|x_n\|=\|x\|$, then if $\lim_{n}x_n=x$?
This question is from problem 6.1.4 in Avner Friedman's "Foundations of ...
- 21
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Rudin's RCA Theorem $4.18$.
There is the definition which we need for the proof:
There is the theorem which we need for the $4.18$:
There is $4.18$:
Let {$u_\alpha : \alpha$ $\in$ $A$} be an orthonormal set in $H$. Each of ...
- 801
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Bounded operator with finite continuous spectrum
Let $T$ be a bounded operator on a complex Hilbert space $H$. Let $\sigma_c(T)$ denote its continuous spectrum. It seems that $\sigma_c$ can in general be quite weird and not "continuous" in ...
- 4,530
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If $A,B\subset H$ are closed subspaces and $\operatorname{codim}(A+B)<\infty$ do we have $A+B$ closed?
Let $H$ be a Hilbert space and $A,B\subset H$ be two closed subspaces such that $\operatorname{codim}(A+B)<\infty $. I would be very surprised if it tuns out that $A+B$ is not necessarily closed. I'...
- 1,001
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40
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Closeness of a set in a infinite dimensional Hibert Space and projection of $0$
I began to read Clarke's Nonsmooth Analysis and the first exercise in chapter is
Let $X$ be a real Hilbert space that admits a countable orthonormal basis $\{e_i\}_{i=1}^\infty$ and set
$$
S := \left\{...
- 23
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1
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Inner-product equality
I am reading Halmos. For some Hilbert space $H$ he has shown $$
\sum_j |(e_i-f_i,f_j)|^2 = ||e_i-f_i||^2.
$$Here $(e_n)$ is an orthonormal basis and $(f_n)$ is some orthnormal set. He directly ...
- 13
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Naming isomorphism of bounded operators on Hilbert spaces coming from isomorphism of Hilbert spaces
Suppose I have an (isometric) isomorphism of separable Hilbert spaces $i:\mathcal{H}_1 \to \mathcal{H}_2$, then I can define $I:\mathcal{B}(\mathcal{H}_1) \to \mathcal{B}(\mathcal{H}_2)$ given by
$$
...
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1
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If $|\omega(\xi)|\le \lambda \|x\xi\|$ for all $\xi \in H$, is $\omega(\xi) = \langle x\xi, \eta\rangle$ for some $\eta \in H$?
Let $H$ be a Hilbert space. Let $\omega \in H^*$ be a bounded functional, $x\in B(H)$ and $\lambda \ge 0$ satisfy
$$|\omega(\xi)| \le \lambda \|x\xi\|$$
for all $\xi \in H$.
Does there exist $\eta \in ...
- 605
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Rudin's RCA, Theorem $4.15$
There is the definition which we need for $4.15$
There is the theorem :
There is $4.15$:
We want to drop the finiteness condition that appears in Theorem $4.14$ without even restricting ourselves ...
- 801
1
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1
answer
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Proving that the domain of the position operator is at least the nuclear space (inequality)
The question:
Consider the function space $\Omega$ which consists of all $\phi(x)$ that satisfy the infinite set of conditions,
$$\int_{-\infty}^{\infty}|\phi(x)|^2(1+|x|^n)<\infty$$
for any $n\in ...
- 725
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1
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36
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Proof of Convergence of the Sum of Components in a Hilbert Space
I recently started studying the fascinating mathematical structures of Hilbert spaces. As a physics guy, I worked with Hilbert spaces in quantum mechanics without knowing the rigorous definition of ...
- 367
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2
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Proving Equality Regarding the Adjoint of Bounded Linear Operator
I am proving the Proposition 2.13 from Elementary Functional Analysis by MacCluer, mainly on c) and d)
We're given that For any $A,B \in \mathscr{B}(\mathscr{H})$, we have
\begin{align*}
(\alpha A)^* &...
- 199
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1
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28
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The proof of $||T|| = \sup\{|(T(f),f)| \;|\; ||f|| = 1 \}$ when $T = T^*$
I'm studying a property of symmetric linear operator in Hilbert space in Stein's Real analysis chapter 4.
When $T = T^*$, then $\Vert T\Vert = \sup\{|(T(f),f)| \;|\; \Vert f\Vert = 1 \}$
The following ...
- 133
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1
answer
34
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linear functional and inner product
Currently, I'm studying the relationship btw linear functional and inner product, and this is the associated theorem in Stein's Real Analysis. (theorem 5.3, chapter 4)
I understand all of the proofs, ...
- 133
1
vote
1
answer
40
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Spectrum of a $T^*T$ for $T$ normal
Let $T$ be a bounded normal operator on a complex Hilbert space $V$. Suppose $\alpha \in \mathbb{C}$ is in the spectrum of $T$, i.e. $T - \alpha I$ is not invertible. I'd like to show that $|\alpha|^2$...
- 178
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1
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40
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Smallest closed subspace which contains $M$
How can one prove following:
If $M\neq \emptyset$ is any subset of Hilbert space $H$, show that $M^ {\perp \perp }$ is the smallest closed subspace of $H$ which contains $M$.
I know that $M \subset M^ ...
- 149
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1
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45
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Pre-Hilbert space and its completion
I'm currently managing to understand how to find a completion of Pre-Hilbert space $\mathcal H_0$ in Stein's real analysis. The textbook says the completion $\mathcal H$ has three properties, and the ...
- 133
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1
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35
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A question on self-adjoint operators
Let $A$ be a unbounded, self-adjoint operator on some Hilbert space $H$. Suppose that $\lVert (\lambda - A) x \rVert \geq \lambda \lVert x \rVert$ for all $x \in D(A)$ and $\lambda > 0$ and let $y \...
- 4,912
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Banach space and Hilbert space
I'm trying to understand Hilbert spaces and Banach spaces, and there is specifically one thing that I don't understand.
As of my understanding the Hilbert space have a scalar product that is defined ...
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