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
58 views

Generalized Poincaré Inequality on H1 proof.

let's see if someone can help me with this proof. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. And let $L^2\left(\Omega\right)$ be the space of equivalence classes of square integrable ...
2
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0answers
21 views

Norm of infinite dimensional Hilbert space to calculate difference between string lengths

I am trying to wrap my head around Proposition 13, last para, page 1049 in this paper. The authors are trying to prove certain properties of string edit distance (defined at the start of Section of 6....
1
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1answer
19 views

A bounded linear functional on a Hilbert space that is a Hahn-Banach extension of one on a subspace

Let $M$ be a closed linear subspace of a Hilbert space $H$ and $g\in M*$(all bounded linear functional on $M$). Let $\pi$ be the orthogonal projection of H onto M, then $f=g\circ\pi$ is the only Hahn-...
2
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1answer
76 views

Exercise 2 , chapter 5 , Stein & Shakarchi real analysis

Consider the Mellin transform defined initially for continuous function $f$ of compact support in $R^+=${$t\in R:t>0$} and $x\in R$ by $Mf(x)=\int_0^\infty f(t)t^{ix-1}dt$ Prove that ($2\pi$)$^{-...
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1answer
15 views

Separable infinite-dimension Hilbert space and its subspaces

Suppose $H$ is any separable infinite-dimensional Hilbert space. Then $H$ has family of closed subspaces $\big\{ E_t :~ t \in [0,1]\big\}$ such that $E_s$ is a strict subspace of $E_t$ for all $0 \leq ...
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1answer
11 views

Weakly square summable series as operators on Hilbert spaces

Let $H$ be a Hilbert space and let $\{a_n\}_{n\in\mathbb{N}}$ be a sequence in $H$ such that $\sum^{\infty}_{n=1}|\langle h,a_n\rangle|^2<\infty$ for all $h\in H$. Here $\langle\dot{},\dot{}\rangle$...
2
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1answer
23 views

GNS-Construction: Involution

Given a C*-algebra $\mathcal{A}$. (It may or may not contain identity!) Consider a positive linear functional: $$\omega:\mathcal{A}\to\mathbb{C}:\quad A\geq0\implies \omega(A)\geq0$$ Construct its ...
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0answers
13 views

References to: If $C\subset\mathbb{R}^n$ is convex and $0\notin C$ then there exists $v\in C$ such that $C$ is in the closed halfspace $H_v$.

For each $v\in\mathbb{R}^n$, we define the notation $H_v=\{u\in\mathbb{R}^n:\langle u,v\rangle\geq0\}$, where $\langle\cdot,\cdot\rangle$ denotes the usual inner product in $\mathbb{R}^n$. Recently, ...
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0answers
15 views

Transformation of inner product of wave functions under transformation of metric

Assume that we have a wave function $\psi(x)$ in the coordinate system $x$ in the Hilbert space $H_1$. The inner product of two states $\psi_1$ and $\psi_2$ are given as $\langle\psi_1|\...
4
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1answer
35 views

A convex subset of a Hilbert space

Assume $C$ is a convex subset of a Hilbert space $H$ ($C$ is not necessarily close) and $x_0\notin C$.Let $r=d(x_0,C)$. Prove that $\{y\in H\mid\|y-x_0\|\leq r\}\cap C$Has at most 1 element. I want ...
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2answers
176 views

Prove of inequality under a Hilbert space.

Let $x\neq y$ when $x,y\in H$ and H is a Hilbert space which satisfy $\|x\|=\|y\|=r$. Show that $\|\frac{x+y}{2}\|<r$. Actually in my question r=1 but as far as i could understand there is a way ...
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1answer
29 views

Can I write $H^1$ as $H^1_0 \oplus H^1_{\perp}$?

Let $\Omega\subset \mathbb{R}^d$, with $d\in \{1,2,3\}$ be an open bounded, simply connected domain. Define $H_0^1$ as the subspace of $H^1$ whose member functions have vanishing trace on the ...
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1answer
32 views

Properties of Injective Operator on Hilbert Space

I am new to functional analysis and have the following issue: Given an infinite dimensional Hilbert space $H$ and an operator $f: H \times \Omega \to H$, where $\Omega$ is some finite dimensional ...
0
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1answer
24 views

Space filling curves: initial definitions

I am confused on the definition of curve and space filling curve in Chapter 1 of the book by Sagan. I think my confusion comes from notation. Let $\mathcal{I}:=[0,1]$, $\mathcal{Q}:=[0,1]^2$ and $J_n$ ...
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1answer
47 views

An expression for the Hilbert-Schmidt inner product

Suppose that $k:[0,1]\times[0,1]\to\mathbb C$ is a Hilbert-Schmidt kernel, i.e. $$ \int_0^1\int_0^1|k(x,y)|^2\mathrm dx\mathrm dy<\infty. $$ The associated Hilbert-Schmidt integral operator $K:L^2([...
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0answers
24 views

Von Neumann algebraic Quantum Object is direct sum of type I factors

I am looking at the non-standard quantum projective spaces $A:=\mathcal{A}_q(\mathbb{CP}^n(c,d))$ introduced by Dijkhuizen and Noumi. Now I want to show that if I take the von Neumann algebra ...
0
votes
1answer
15 views

Product of Lebesgue measure on Hilbert cube doesn't satisfy doubling condition?

The Hilbert cube $H$, is the infinite dimensional product $[0,1]\times [0,\frac12]\times...$ Let $\mu$ be product of Lebesgue measures $\mathcal{L}^1 \times \mathcal{L}^1\times...$, I heard that the ...
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1answer
23 views

Functionals taking real values

Suppose $f$ is a bounded functional on a separable Hilbert space. Can we always find an orthonormal basis such that $f$ takes real values on that basis?
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0answers
17 views

How to prove that $ e_{\lambda}$ can be written in the following form?

Let $e_{\lambda}$ be the spectral density associated to the spectral function $E_{\lambda}$ for a self-adjoint operator $A$ on a complex Hilbert space $(H,\left<., .\right>)$. Haw to prove that ...
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0answers
16 views

Question on Hermite functions

Stein - Real Analysis p.205 Hermite functions $h_k(x)$ are defined by the generating identity $\sum_{k=0}^\infty h_k(x)\frac{t^k}{k!} = e^{-x^2/2 + 2tx - t^2}$. I have proven that it satisfies ...
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1answer
27 views

norm from inner product

I have a question in a Hilbert Spaces course as follows: Let $X=(x_1, x_2)$ be vector in a vector space of all ordered pairs of complex numbers X. Can we obtain the norm defined on X by: $\|X\|=|x_1|...
0
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1answer
33 views

unitary operator between two Hilbert subspaces

$H$ is a Hilbert space. $P, Q$ are projections. For every $x\in P(H)$, we have decomposition $x = Qx +Q^\perp x$. Then, can we find a unitary operator from the space generated by all $Qx$, $x\in P(H)$...
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2answers
41 views

Norms with complex numbers over Hilbert Spaces

Let $H$ be a Hilbert space and $v,w \in H$ ans a be a scalar. Prove that $\|v\| \leq \|v+aw\|$ for all scalar a iff (v,w)=0 for real and complex cases. I want to choose a such that $\bar{a}(v,w)$ ...
2
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1answer
44 views

projections in von Neuman algebra

Consider a semifinite von Neumann algebra $\mathcal{M}$ with a semifinite faithful normal trace $\tau$. If $Q, P$ are projections in $\mathcal{M}$ with $\tau(Q)< \tau(P)$, then does $\tau(P\wedge Q^...
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1answer
23 views

Property of Conical Hull

Let $H$ be a real Hilbert space and $C$ be a nonempty convex subset of $H$. The conical hull of $C$ is defined by $$ \operatorname{cone}{C} := \bigcup_{\lambda >0}{\lambda C}. $$ (it is a cone in ...
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0answers
9 views

Bounded input-Bounded output stability for countable system of ODES.

Let $X$ be a countably infinite dimensional Hilbert space. Let $f\colon X\to X$ be a compact, linear, symmetric positive definite map. Define an ODE as $u_t = -f(u-y)$ and $u(0) = 0$, where $y\...
2
votes
1answer
69 views

How can we prove that the space of trace class operators on a Hilbert space $H$ is the closure of $H\otimes H$ with respect to the trace norm?

Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space over $\mathbb R$ $\mathfrak L^1(H)$ be the space of trace class operators on $H$ and $$\operatorname{tr}L:=\sum_{n\in\mathbb ...
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0answers
36 views

Is there a name for these inequalities? Where can I look them up?

Consider the operators $A,B,C$ on Hilbert space $\mathcal H$: Show that: $$ \left \vert \left \vert AB \right \vert \right \vert \le \left \vert \left \vert A\right \vert \right \vert \left \vert \...
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votes
1answer
82 views

A question concerning Mazur's Lemma

I have a problem with application of Mazur's Lemma. Just consider $B(H)$ when $H=\ell_2$. Then, $B(H)$ is a normed vector space. Then, take operators $$X_n:={\rm diag}(0,0,\cdots,0,1,1,1,1,\cdots)$$...
0
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1answer
16 views

Almost negative definite matrices and norm-distance matrices

An "almost negative definite" matrix $A$ satisfies the property $$ v^te = 0\implies v^tAv\le 0 $$ where $e=(1,1,\dots,1)$. We know that if $A$ is a simmetric zero-diagonal (hollow) matrix, then $A$ ...
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0answers
28 views

Metrics from Operator Norms

Let $X$ be a Hilbert space and $(\cdot,\cdot)_X$ be the inner product on $X$. It is well known that $|x|_X = \sqrt{(x,x)_X}$ is a norm on $X$ and $|x-y|_X$ is a metric on $X$. The norm on $X$ induces ...
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1answer
31 views

Showing that the intersection of two closed linear subspaces is the trivial subspace.

I'd appreciate if someone can provide the best way to deal with this problem. Let $\{\alpha_n\}$ be an orthonormal sequence for a Hilbert space H and let $\{\beta_n\}$ be an orthonormal sequence such ...
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2answers
48 views

Norm of operator $A$ st. $A^2 = I$?

I'm wondering what can be said about the norm $||A||$ of an operator which squares to identity. All I can think of is that $$1=||AA|| \leq ||A||^2$$ so that $||A|| \geq 1$. But can anything else be ...
2
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1answer
30 views

Unique ground state of Schrödinger Operators

I'm reading a book and there is an argument that the ground state of a Schrödinger operator is unique. The problem is I think the argument is complete non-sense! These are lecture notes by Witten, I ...
2
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1answer
39 views

Weak convergence and strong convergence on $B(H)$

Let $\mathcal{A} \subset B(H)$ be a weak closed convex bounded set of self-adjoint operators. If $A_n \rightarrow_{wo} A\in \mathcal{A}$, do we have $A_n \rightarrow A$ strongly?($A_n$ is a sequence ...
3
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1answer
26 views

If operators $A, B, A+B$ are all closable, show that $\overline{A + B} \supseteq \overline{A} + \overline{B}$.

Let $A, B$ be closable, unbounded, linear operators on a Hilbert space $H$. Suppose further that the operator $A + B$, defined on the intersection of domains $D(A) \cap D(B)$, is also closable. I ...
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1answer
23 views

the orthogonal complement intersection of sets

Let $\{C_\gamma\}$ be a net of subsets of a Hilbert space(or some other spaces). Do we have $$(\wedge_\gamma C_\gamma )^\perp = \vee_\gamma C_\gamma ^\perp?$$ It is known that it works when $\{C_\...
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7answers
5k views

What is “Bra” and “Ket” notation and how does it relate to Hilbert spaces?

This is my first semester of quantum mechanics and higher mathematics and I am completely lost. I have tried to find help at my university, browsed similar questions on this site, looked at my ...
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0answers
37 views

Spectrum of Laplace operator with potential acting on $L^2(\mathbb R)$ is discrete

Consider an operator $H=-\Delta +U(x)$ on $L^2(\mathbb R)$ for a function $U(x): \mathbb R \to \mathbb R$ that tends to $+\infty$ as $|x|$ grows. These kinds of operators appear all over non-...
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1answer
18 views

Show a function is Sobolev

Let $T_{h}$ be a subdivision of a domain $\Omega \subset \mathbb{R}^d$ into elements $K$ with boundary $\delta K$so that the Gauss divergence theorem holds. If for a function $f$ it holds that $f \...
2
votes
1answer
29 views

Tomita Theory: Involution

Given a Hilbert space $\mathcal{H}$. Consider a von Neumann algebra: $$M\subseteq\mathcal{B}(\mathcal{H}):\quad M=M''$$ Suppose a cyclic vector: $$\Omega\in\mathcal{H}:\quad\overline{\mathcal{M}\...
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2answers
46 views

Function $f$ such that $f''\in L^2(\mathbb{R})$

Let us assume that $f\in L^2(\mathbb{R})$ and $f''\in L^2(\mathbb{R})$ ($f'$ - first derivative,$f''$ - second derivative), i.e. $f$ is square-integrable, $f$ is differentiable, its first deriviative ...
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0answers
18 views

Given an operator $Q$ between a Hilbert space $U$ and $L^2(ℝ^d;ℝ^d)$, is it possible to make sense of $U∋u↦(Qu)(x)$ for a fixed $x∈ℝ^d$?

Let $U$ be a Hilbert space $H:=L^2(\mathbb R^d;\mathbb R^d)$ for some $d\in\left\{2,3\right\}$ $Q$ be a Hilbert-Schmidt operator from $U$ to $H$. I want that $\tilde Q(x)$, where $$\tilde Q(x):=...
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0answers
17 views

Extending a unitary isomorphism on a Hilbert space

Let $H$ be a Hilbert sapce and $M$ a dense subspace of $H$. Prove that any unitary isomorphism on $M$ can be uniquely extended to a unitary isomorphism on $H$. So here's what I tried: Let $T: M \...
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0answers
18 views

Characterizing when kernels in Reproducing Kernel Hilbert Space (RKHS) are linearly independent

In my studies of RKHS i.e. Reproducing Kernel Hilbert Spaces stating the following Let $ \mathbb{H} $ be a RKHS on a set X. We are asked to characterize when the following set of kernels $ \{ k_{...
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1answer
18 views

some detail calculation on the proof of equivalence of norms

We say that two norm $\|x\|_1$ and $\|x\|_2$ on a vector space $X$ are said to be equivalent if there exists $K>0$ and $M>0$ such that $$ K\|x\|_1\le \|x\|_2\le M\|x\|_1 $$ Prove that on a ...
3
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3answers
76 views

Why are function spaces generally infinite dimensional

The other day, I was trying to explain some concepts in Fourier analysis and wavelets to my girlfriend (an electrical engineering student) and obviously, the concept of Lebesgue integration came up in ...
0
votes
1answer
48 views

Approximate point spectrum of a normal operator

how can I show the following theorem? Let $H$ a Hilbert space and $T:H \to H$ a linear, continuos and normal operator. Then for every $\lambda \in \sigma(T)$ there exists a sequence $(x_n)_{n \in \...
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1answer
25 views

Spectral projections, additivity

Let $K$ be a positive operator on a Hilbert space $H$. $Q_1$ and $Q_2$ are projections such that $Q_1\perp Q_2$. Is $$ E^{Q_1K Q_1} (1,\infty) + E^{Q_2K Q_2} (1,\infty) =E^{Q_1K Q_1 +Q_2K Q_2} (1,\...
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0answers
20 views

Two definitions of the operator $\exp(x)$ in $L^2(\mathbb R)$

The operator $x$ acts on a dense subspace of $L^2(\mathbb R)$ and is not bounded. So if we define $\exp(x)$ via the power series $\sum_{n=0}^\infty \frac {x^n}{n!}$, convergence will not follow in the ...