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Questions tagged [hilbert-spaces]

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

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The completion of $F/\text{Ker}(M)$ is isomorphic to $(\overline{\text{Im}(M)}, \langle\cdot,\cdot\rangle)$

Let $M$ be a positive semidefinite operator on a complex Hilbert space $(F,\langle\cdot,\cdot\rangle)$. On the quotient space $F/\text{Ker}(M)$ we have the following inner product $$( \overline{x},\...
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2answers
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$PQ = 0 \Leftrightarrow P(H) \bot Q(H)$ if $H$ is Hilbert and $P,Q$ orthogonal projections.

Exercise : Let $H$ be a Hilbert space and $P,Q \in \mathcal{L}(H)$ orthogonal projections. Show that : $$PQ = 0 \Leftrightarrow P(H) \bot Q(H)$$ Attempt-Thoughts : $(\Rightarrow)$ Let $PQ = 0$. ...
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1answer
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$PQ \; \text{orthogonal projection} \; \Leftrightarrow PQ = QP$

Exercise : Let $H$ be a Hilbert space and $P,Q \in \mathcal{L}(H)$ are orthogonal projections, then show that : $$PQ \; \text{orthogonal projection} \; \Leftrightarrow PQ = QP$$ Seeking a formal ...
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25 views

Operator identity in Quantum mechanics

The question is : If an operator $\hat{A}$ follows the property that $\hat{A}.\hat{A}=\mathbb{I}$ , where $\mathbb{I}$ is the identity operator, then prove that: $\exp(\theta \hat{A})=\cosh (\...
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Testing if cross-covariance is zero in Hilbert spaces

I'm attempting to generalize a multivariate test of zero cross-covariance between two random varaibles to infinite dimensional Hilbert spaces and I'm looking for some advice / ideas on how to work ...
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2answers
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Continuity of Linear Operator Between Hilbert Spaces

Note: Please do not give a solution; I am curious to understand why my solution is incorrect, and would prefer guidance to help me complete the question myself. Thank you. Let $\mathcal{H}$ be a ...
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36 views

Finite rank operators on Hilbert spaces

Let $H$ be a Hilbert space. Question 1: Are all rank one operators from $H$ to $H$ is of the form $$T:H\rightarrow H, x \mapsto \langle x,u\rangle v $$ For some $u,v \in H$. Question 2:...
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A question on the Non-degenerated bilinear form

Prove that $<C(G), C(G)>$ is dual system with bilinear form $$\int_G \phi(x)\psi(x)dx $$ and $\psi(x),\psi(x)\in C(G)$ so here i was trying bilinear form im not getting how to prove non-...
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1answer
23 views

Convergence of finite dimensional projection of trace class in trace norm

Assume $\mathbb{H}$ is a Hilbert space and $K$ is a trace-class operator on it. Given a fixed ONB $\{e_i\}$ and assume $$K=\sum_{i,j}c_{ij}e_i\otimes e_j.$$ Now, let $K_n = \sum_{1\leq i,j\leq n}c_{...
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Non-Negative Vs Positive Semi Definite

A matrix is PSD if $$\langle Ax, x\rangle \ge 0, \forall x \in H$$ Where, H is a hilbert space and A is a mapping $H \rightarrow H$. Is it the same as being Non-negative? I couldn't seem to find a ...
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2answers
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If $A \in \mathcal{L}(H)$ and $\langle A(u),u \rangle \geq \langle u, u \rangle$, then $A$ is invertible.

Exercise : Let $H$ be a Hilbert space and $A \in \mathcal{L}(H)$ such that : $$\langle A(u),u \rangle \geq \langle u, u \rangle \; \forall u \in H$$ Show that $A$ is invertible. Attempt/...
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25 views

Show that graph of operator with adjoint operator is closed

Let $X,Y$ be inner-product spaces. Let $T\in L\left(X,Y\right)$ be a linear operator with adjoint operator $S\in L\left(Y,X\right)$ such that $$\langle Tx,y\rangle_Y=\langle x,Sy\rangle_X\quad\forall (...
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Density of a subspace

Let $\Omega\subset\mathbb{R}^d$ be open, bounded and simply connected with smooth boundary $\partial\Omega$. Define $\mathcal{H}:=\{u\in H^1(\Omega)~|~\Delta u\in L^2(\Omega)\}$ with norm $$\|u\|_\...
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3answers
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Examples of non-unitary isometries on finite dimensional Hilbert spaces?

I was reading the question A Finite Dimensional non-Unitary Isometry?, which gives an example of a non unitary isometry which is a map $T: R \rightarrow R^2 $. This question is based on a previous ...
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1answer
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Why is a finite metric spaces is not always of negative type?

A finite metric space $(X,d)$ is of negative type if $d$ satisfies $$ \sum_{x\in X} c_x=0 \implies \sum_{x,y\in X} c_x c_y d(x,y)\leq 0 $$ - def from [M. DEZA AND H. MAEHARA, METRIC TRANSFORMS AND ...
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How to find explicit feature map $\phi $? [closed]

Given, polynomial kernel K(x,y) = $(x^Ty)^2$ for x,y $ \in R^2$ and $< \phi (x), \phi (y)> = K(x,y)$ for $ \phi : R^2 -> R^3 $ where the inner product the standard inner product in $R^3 $
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1answer
17 views

Sufficient condition to show density of $\mathcal{E}$ in $\Lambda^2$(Hilbert space)

I am trying to understand the proof that the space of simple processes is dense in $\Lambda^2$. The proof in my lecture notes starts by assuming that for $\phi \in \Lambda^2$ which is orthogonal to ...
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1answer
23 views

Improper integral convergence for different powers of a fraction

I read a solution for some exercise in Hilbert spaces theory, which included some improper integrals convergence which I did not compeletly understand: If it matters, the original question was - ...
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2answers
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Let $a, b$, be positives and $a \leq b$. For which $p$ the function $\frac{1} {x^a+x^b}$ is $L_p(0,+\infty)$?

Let $a, b$, be positives and $a \leq b$. For which $p$ the function $\frac{1} {x^a+x^b}$ is $L_p(0,+\infty)$? I have solved without any problem the case $p=\infty$ (and it is not $L_p$ in this case) ,...
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How do I show completeness of a trigonometric orthonormal system?

Consider the system $$ T = \{ 1, \cos(x), \sin(x), \dots, \cos(nx), \sin(nx), \dots \} $$ I can show that they form a orthonormal system on $L^2( [ -\pi, \pi ] )$, but I don't know how to show how ...
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1answer
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Given $X$ Hilbert space, $T\in X^*$, $y$ projection of $x_0$ on $Y=\text{Ker}T$, why does $x-\frac{T(x)}{T(x_0-y)}(x_0-y)\in \text{Ker}T$?

Let $X$ Hilbert space $T\in X^*$, ie $T:X\to\mathbb R$ linear $Y=\text{Ker}T=\{x\in X:T(x)=0\}$ closed subspace of $X$ $x_0\in X\setminus Y$ $y\in Y$ orthogonal projection of $x_0$ on $...
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1answer
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Does Pythagorean law holds in Hilbert space or plane?

Does Pythagorean law holds in Hilbert space or plane? I know that Pythagorean law holds in Euclidean plane while Hilbert space is a generalisation of Euclidean space. I know that finite orthonormal ...
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Can anyone prove this proposition? (Frame Functions)

Can anyone prove this proposition? It is proposition 3.2.2. out of A. Dvurecenskij's book on the applications of Gleason's Theorem. The proof in the book just says that it is obvious and doesn't ...
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2answers
53 views

No non-negative continuous function on $[a,b]$ such that $\int_a^b f(t)dt=1, \int_a^b tf(t)dt=c, \int_a^b t^2f(t)dt=c^2$ for $c\in\mathbb{R}$.

Show that there exists no non-negative continuous function $f$ defined on the interval $[a,b]$ such that it satisfies the following conditions: $$\int_a^b f(t)dt=1 \quad \int_a^b tf(t)dt = c \quad \...
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Condition on a diagonal operator to be *-cyclic

Consider a diagonal operator $T: l^2 \to l^2$. What do we need (condition on $T$) for $l^2$ to be a cyclic $C(\sigma(T))$-module? In other words when there exists $x \in l^2$ such that $C(\sigma(T)) \...
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Can I equip the space $H^{1/2}(\partial \Omega)$ with a $L^2( \partial \Omega)$ inner product?

Since the space $H^{1/2}(\partial \Omega)$ with its norm $\Vert u \Vert_{H^{1/2}(\partial \Omega)} = \inf_{v \in H^1(\Omega), v\vert_{\partial \Omega} = u} \Vert v \Vert_{H^1(\Omega)}$ gets unhandy ...
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Factoring Variable in $L^2$

Suppose that $X,Y$ are random-variables in $L^2(\Omega,\mathcal{F},\mathbb{P})$, for some complete probability space $(\Omega,\mathcal{F},\mathbb{P})$, and suppose that there exists a Borel function $...
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1answer
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Is the unitary group connected in infinite dimensions?

Let $H$ be an infinite dimensional Hilbertspace and $U(H)$ the group of unitaries endowed with the norm topology. Is $U(H)$ connected? The following is a generalisation of the proof in the finite-...
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1answer
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Orthogonal decomposition with a special inner product

Assume that we are working on $\mathbb{R}^p$ with an inner product induced by the positive definite matrix $\mathbf{G}$, i.e. for $\mathbf{f}, \mathbf{g} \in \mathbb{R}^p$ we define $\langle \mathbf{f}...
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1answer
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Proving that $\|Av\|\geq \lvert \langle u, v\rangle\rvert\cdot \|A\|$ for $\|Au\| = \|A\|$ in a Hilbert space

I've shown that for a matrix $A\in \mathbb{R}^{n\times n}$ and an arbitrary $v\in \mathbb{R}^n$, we have the inequality $$\|Av\|\geq \lvert \langle u, v\rangle\rvert\cdot \|A\|$$ where $\|\cdot\|$ is ...
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How to handle the following problem in Hilbert spaces over complex numbers.

Let $k = [k_{ij}]_{i, j=1}^{\infty}$ be an infinite matrix over $\mathbb{C}$ (the set of all complex numbers) such that (i) for each $i \in \mathbb{N}$ (the set of all natural numbers), the $i$th row ...
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0answers
50 views

Proof of why $S^\infty$ is contractible using Hilbert space

In John Baez's blog post http://www.math.ucr.edu/home/baez/week151.html, Baez gives a short proof of why $S^\infty$ is contractible, using that $S^\infty$ is the unit sphere in a infinite dimensional ...
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1answer
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Pre-Hilbert spaces and the Riesz Representation Theorem.

I'm looking for an example of a Hilbert space $(H,\langle \cdot,\cdot\rangle)$ that satisfies the following: In $H$ there exists an element $a$ such that $(H \backslash \{a\},\langle \cdot,\cdot\...
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Reconstructing a measure space from an $L^2$ space.

Consider a measure space $(X, \mathcal{B}, \mu)$. By quotienting out its null sets, we get the measure algebra $(\tilde{\mathcal{B}}, \tilde\mu )$. By considering square-integrable functions out of $X$...
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1answer
24 views

$u_j \rightarrow u$ and $||Tu_j||$; exists sequence $w_k$ such that $Tw_k $ converge in norm.

Let $H$ be a Hilbert space. Let $T:dom(T) \rightarrow H$ be a densely defined operator. Suppose there is a sequence $\{u_j \}$ in the domain of $T$ such that $u_j \rightarrow u$, and $||Tu_j||$ is ...
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Integration in first Sobolev space

Let be $H^1(R)$ the first Sobolev space: $$H^1(R)=\{f\in L^2(R)\mid \quad f'\in L^2(R) \quad \}$$ where the derivative is intended in distributional sense. I have to show that the following equations ...
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what does it mean? "the function class becomes an approximation to the Reproducing Kernel Hilbert Space corresponding to the Gaussian kernel.

It seems like embedding. But I am not sure what approximation to RKHS means. Or how to prove something is approximation to RKHS?
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How to prove that a subspace of Hilbert space is also a Hilbert space? [closed]

I have problem with proving that the subspace of Hilbert space is still a Hilbert space.
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1answer
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Orthogonal projection onto the closure of a subspace spanned by a set of orthonormal vectors

Suppose $H$ is a Hilbert space and $\{e_1,\cdots\}$ is a set of infinitely many orthonormal vectors. Then let $M$ be the closure of the subspace spanned by these orthonormal vectors. I know that for $...
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1answer
34 views

Projection onto the range of an operator in a nonseparable Hilbert space

Let $A$ be a matrix with linearly-independent columns. Then $A(A^TA)^{-1}A^T$ is the orthogonal projection matrix onto the range/image of $A$. This formula is legal because the linear independence of $...
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2answers
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Null Space of Bounded Linear Functionals

Assume $f$ is a linear bounded functional from $\mathbb{R}^2\rightarrow\mathbb{R}$. I understand that the null space of $f$, denoted $\mathcal{N}(f)$, is a subspace of $\mathbb{R}^2$. However, I am ...
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0answers
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Is there any incompatibility between affine spaces and Hilbert spaces? [closed]

I was wondering if there is such a thing like a Hilbertian affine space. I've seen the definition of an Euclidian affine space, which is: An affine space (A, V, φ) is an Euclidean affine space if ...
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Show {$x^k$}$_{k\geq 0}$ is complete in $L^2[a,b]$

I'm thinking of using the Weierstrass Approximation Theorem where the span of monomials is dense in the continuous functions.
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Norm of Hilbert space Gaussian

Let $X$ be a random variable with values in a separable Hilbert space $\mathcal{H}$ with inner product $\langle \cdot, \cdot \rangle$. Assume that $X$ has a Gaussian distribution in the sense that $$...
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0answers
35 views

If $X$ is a vector space endowed with a metric that is NOT norm induced and If $x_n\to x$ and $y_n\to y$, then $x_n+y_n\to x+y$

Let $X$ be a real or complex vector space endowed with a metric $d$ which is not induced by a norm, that is there exist no norm $\| \cdot \|$ on $X$ such that $d(x,y)=\| x-y \|$. Prove or disprove: ...
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1answer
41 views

Is this multiplication operator bounded for this special norm?

Consider $L_2[0,1]$ with the usual inner product $\langle f, g \rangle = \int_{0}^1 f(t)g(t) \, dt$ and define a new norm $$ \| f \|^2_{\star} = \sum_{i=1}^\infty \langle f, \phi_i\rangle^2 \lambda_i ...
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1answer
29 views

Operator norm of integral operator

Suppose we have $X=L^2([0,1];\mathbb{R})$ and \begin{equation} T:X\rightarrow X, \ Tf(x)=\int_0^1x^2yf(y)dy. \end{equation} Show that $T$ is compact and determine $||T||.$ I already have that $||T|...
2
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1answer
41 views

Showing that $\exists x \in H : \|A(x)\| = \|A\|_\mathcal{L}$ if $H$ is Hilbert and $A \in \mathcal{L}_c(X,Y)$.

Exercise : Let $H$ be a Hilbert space and $A \in \mathcal{L}_c(H)$. Show that $\exists x \in H : \|A(x)\| = \|A\|_\mathcal{L}$. Attempt - Thoughts : Note : The space $\mathcal{L}_c(H)$ is the ...
1
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1answer
28 views

Interior solution of a Variational Inequality in Hilbert Space

I'm trying to understand the proof of the following claim: Let $K \subset \mathbb{R}^n$ be compact and convex and let $$F:K \rightarrow (\mathbb{R}^n)'$$ be continuous. Then there exists an $x\in K$...
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1answer
32 views

Chapter 3 Problem 23 Luenberger Optimization by vector space methods

I have some problem with problem 3.23 in Luenberger book "Optimization by vector space methods". Consider the problem of finding the vector $x$ of minimum norm satisfying $(x|y_i) \geq c_i, i=1,2,\...