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

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1answer
30 views

Operator with norm

I got the following problem to solve: Let $H$ Hilbert space and $T: H \to H$ a bounded positive operator, i.e. \begin{align*} \langle x, T x \rangle \geq 0 & & \text{for all } x \in H. ...
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2answers
21 views

Showing $A_{ij} = (Ae_i, e_j)$ for matrix $A$ of complex linear operator $\mathbb{C}^n \to \mathbb{C}^n$ and orthonormal basis $(e_i)_{i=1}^{n}$

I'm sure this is a simple question, but I get stuck in the algebra when I try to prove it from definition. Suppose $A$ is an $n \times n$ matrix representing a complex linear operator $\mathbb{C}^n ...
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1answer
17 views

How to show that this integral operator is bounded?

Consider the integral operator $T : C([0,1])\to C([0,1])$ given by $$Tf(t)=\int_0^1 K(t,\tau)f(\tau)d\tau.$$ I'm solving one exercise which is to show this operator is bounded. The exercise is from ...
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1answer
29 views

Isometry on Hilbert space [on hold]

If $T: H \rightarrow H$ be a function on hilbert space $H$ that satisfies $(Tx | Ty) = (x |y)$ for all $x$ and $y$. I need to show that $T$ is linear and an isometry in $B(H)$.
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1answer
21 views

Bounded operators on inner products on Hilbert space

If we have a Hilbert space $H$ with inner product $( \cdot | \cdot)$, and let $( \cdot| \cdot)_1 $ be another inner product on $H$ such that $(x | x)_1 \leq (x | x)$ for every $x \in H$. I was trying ...
2
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1answer
75 views

Prob. 9, Sec. 3.10 in Kreyszig's functional analysis book: The image of ann isometric non-unitary operator on a Hilbert space

Let $H$ be a Hilbert space, let $T \colon H \to H$ be a linear operator such that $T$ is isometric but not unitary. Then how to show that the image $T[H]$ is a proper closed subspace of $H$? My ...
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1answer
33 views

Proof about orthogonality of columns of a matrix

Consider a matrix $A \in \mathbb{R}^{n \times n}$ and the canonical inner product in $\mathbb{R}^{n}$. Show that if the rows of A form an orthogonal set, the same happens with the columns. So ...
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 ...
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0answers
20 views

Hausdorff-Quotient: Embedding

Problem Given a uniform space $\Omega$. (Exemplary Topological Vector Space!) Consider a dense subspace: $$\iota:\mathcal{D}\hookrightarrow\Omega:\quad\overline{\iota\mathcal{D}}=\Omega$$ Regard ...
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1answer
17 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 ...
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1answer
53 views

Bounded operator on Hilbert space

Let $H$ is a Hilbert space. If $T\in B(H)$ show that $T+T^*\ge 0$ iff $T+I$ is invertible in $B(H)$ with $\|(T-I)(T+I)^{-1}\|\le 1$. (Hint is $T+T^*\ge 0$ iff $\|(T+I)x\|\ge \|x\|\ $ and ...
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0answers
36 views

Finding the closest function to another in a Hilbert space.

Let H be the Hilbert space L$^2$([0,1)], and let $S$ be the subspace of functions f $\in$ H satisfying $\int^1_0(1+x)f(x)dx=0$. Find the element of $S$ closest to the function $g\in H$ defined ...
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1answer
49 views

Cardinality of a Hilbert space

I have seen the theorem about the cardinality of orthonormal basis of a Hilbert space. I wonder if we have a Hilbert space $H$ with an orthonormal basis having cardinality of the continuum, then what ...
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1answer
62 views

Minimal projections and Type II von Neumann Algebras.

Let $M \subseteq B(H)$ be a type $II_1$ factor. Can it contain a minimal projection? If it can't, what would go wrong? I assume something about the trace being faithful?
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1answer
123 views

Unique trace on a type $II_1$ von Neumann Algebra

Let $M \subseteq B(H)$ be a type $II_1$ von Neumann Algebra. Then any two non-zero ultraweakly continuous normalised traces $Tr,tr : \rightarrow \mathbb{C}$ are equal. I'm trying to understand this ...
2
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1answer
23 views

Convexity of Hilbert cube [on hold]

I am trying to show that the Hilbert cube $\{ x_n \in l^2(\mathbb{N}) \mid x_n \in [0, \frac{1}{n}] \ \forall n \in \mathbb{N} \}$ is convex and (norm)-compact.
2
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1answer
49 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 ...
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0answers
19 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 ...
2
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1answer
70 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 ...
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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 ...
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1answer
9 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 ...
2
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1answer
21 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
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 ...
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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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1answer
27 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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2answers
170 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
32 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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1answer
77 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 ...
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1answer
25 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 ...
2
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2answers
111 views

Dual space of a closed subspace of a Hilbert space

I'm reading Girault and Raviart's book concerning Finite Element Methods for Navier-Stokes equations, and they use in the proof of one result, the following argument: As $V=\{v\in H_0^1(\Omega)^N; ...
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1answer
23 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
45 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 ...
0
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1answer
13 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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0answers
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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 ...
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0answers
16 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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1answer
22 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
15 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
53 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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1answer
22 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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1answer
25 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: ...
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1answer
25 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 ...
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2answers
40 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 $\bar{a}(v,w)$ ...
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0answers
32 views

Adjoint of differential operator

I would like to find the adjoint of the operator $T_a$ ($a\in \mathbb{C}$) on $ \mathcal{H}=L^{2}(\mathbb{R}^{2},dxdy)$ with $(u,v)=\int \int u(x,y)\overline{v(x,y)} dx dy$ ...
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0answers
7 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 ...
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1answer
14 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
35 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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7answers
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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
24 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 ...