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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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Mean square convergence and mean square integral

Let $\{X(t)|t\in T\}$ be a stochastic process in continuous time with $X(t)\in L_2(\Omega,\mathcal{F},\mathbf{P})$ for all $t\in T$. Its mean square integral $\int_a^b X(t)dt$ over $[a,b]\subseteq T$ ...
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Inner product on a sequence and its limit

I am stuck on a question, and it seems like I'm missing a really obvious Cauchy-Schwarz application or something, but I am left scratching my head. Let $(x_n):n \in \mathbb{N}$ be a sequence in a ...
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$H$ is a Hilbert space , then each orthonormal basis of $H$ has the same cardinality

$H$ is a Hilbert space , $M_0$ and $M_1$ are two orthonormal basis of $H$ , can we prove that $M_0$ and $M_1$ have the same cardinality ? My attempt : If $M_0$ is infinite and countable $M_1$ is ...
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Convergence in a Hilbert Space

I have a homework question I am attempting to no avail. Let $\mathcal{H}$ be a Hilbert space. Let $X\subseteq\mathcal{H}$ be a convex set. Suppose that $(x_{n})_{n\geq 1}$ is a sequence in $X$ such ...
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Is it possible for a solution of a nonlinear differential equation to be written as a sum of linear basis functions in a Hilbert Space?

On the one hand, I think that there are nonlinear differential equations where you can show the solution is not linear. However, I think that solutions to differential equations must be contained in a ...
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Show that $A:\ell^2(\mathbb N)\to\ell^2(\mathbb N)$ where $A(e_n)=\lambda_ne_n$ is bounded.

Let $C\subset\mathbb C$ be closed. As $\mathbb C$ is separable then so too is the subset $C$. This means that there exists a countable subset $\{\lambda_n:n\in\mathbb N\}\subset C$ dense in $C$. In ...
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Is the “tensoring map” from $\mathcal{H}_1 \times \mathcal{H}_2$ to $\mathcal{H}_1 \otimes \mathcal{H}_2$ a continuous map?

Let $\mathcal{H}_1$ and $\mathcal{H_2}$ two Hilbert spaces. Construct the tensor product $\mathcal{H}_1 \otimes \mathcal{H}_2$ as the set of all bounded antilinear operators from $\mathcal{H_2}$ to $\...
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$l_2(S)$is a hilbert space where S is a subset

Let S be a non-empty set and $l_2(S)$ be the set of all complex functions $f$ defined on $S$ with the following two properties: $(1) \{s:f(s)\ne0\}$ is empty or countable. $(2) \sum{|f(s)|}^2 <...
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If an operator $T$ satisfy a property, then $\|Tx\|=c\|x\|$

Let $(E,\langle\cdot\;,\;\cdot\rangle)$ be a complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all operators on $E$. Assume that $T\in \mathcal{L}(E)$ and satisfy the following property (P)...
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Several Questions about Hilbert Manifold

These should be quite fundamental questions about the definitions about Hilbert manifold yet I cannot find answers about them after a long-time search: How is the tangent space at a point on a ...
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Norm of Fourier series

I am reading the proof of the statement that no non-zero multiplication operator on $L^2([0,1])$ is compact in this post. And I would like to address it as a seperate post as I am only curious about ...
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The importance of estimates of frame bounds.

A theorem contained in Christensen, Ole (1995), "A Paley-Wiener theorem for frames." Proceedings of the American Mathematical Society, 123, 2199-2201. states that Let $\{x_n\}_{n\in\mathbb Z}...
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orthonormal basis of infinite dimensional Hilbert space H is not a basis of H as vector space?

Apparently the orthonormal basis $(e_n)_{n\in \mathbb{N}}$ of the Hilbert space $H$ (in special case, infinitly dimensional) is not a basis of $H$ as a vectorspace. Is there a way to prove this?
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Counting Balls in $L^2_m[0,1]$

Setup and Thoughts to Date Let $B$ be the closed unit ball in ${L}^2_m[0,1]$, where $m$ is the Lebesgue measure, equipped with the weak topology. I know that $B$ is separable, so there exists a ...
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Duality map in Hilbert space

Let $X$ be a Hilbert space. Then we have the identification between $X$ and $X^*$ definied by $x=|x| grad|x|$. How do we get that identification?
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Question about inner product of $C_{2}[a,b]$.

My question, I think, is quite simple. I have a space $C_{2}[a,b]$ of complex-valued continuous functions. I checked that functional $$(f,g) = \int_{a}^{b} f(t)\overline{g}(t)dt$$ is a inner product ...
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Find projection-valued measure associated with parity operator

Let's define parity operator as follows: $$\pi:L^2(\mathbb{R})\to L^2(\mathbb{R})$$ $$\psi(x)\mapsto \psi(-x)$$ It's easy to show that $\pi$ is a self-adjoint operator and its spectrum is just $\sigma(...
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Trace of a matrix exponential with tensor products, and Von Neumann entropy

$\def\T{\operatorname{Tr}}$ $\def\1{\mathbb{1}}$ Let $H=H_1\otimes H_2\otimes H_3$ be a finite dimensional Hilbert space, and let $\rho_{123}$ be a self-adjoint matrix with $\rho_{123}\geq 0$ (...
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Integral representation of log of operators

$\def\1{\mathbb{1}}$ Suppose we're in a "good enough" (finite for example) space, and we have positive (semi)-definite operators $P$ and $Q$. Let $\log{(P)}$ and $\log{(P)}$ be logarithms of $P$ and ...
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Converges in inner product space

Assume $(f_i)_{i\in I}$ is an orthonormal/orthogonal system in an (complex) inner product space. Does $$\sum_{i\in I}\langle f_i,f\rangle f_i$$ always converges for any $f$ (may not to $f$)? ...
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Understanding the defintion of dual operators

I'm reading a book about Functional Analysis, and now I've reached to the part about dual operators. I'm having some difficulties understanding the following definition - Why $A^*$ is $Y^*\...
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Compact operator on $L_2([0,1],m)$

Consider the Hilbert space $H=L_2([0,1],m)$ where $m$ is the Lebesgue measure on the interval $[0,1]$. Let $T \in \mathcal{L}(H,H)$ given by \begin{equation*} T\ f(x)=x \ f(x) \ \ \ \ f \in H,\ x \...
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Show that this is an inner product

Let's define $$(f,g)=\int_{\mathbb{R}} \frac{f(x)\bar{g}(x)}{1+x^2}dx$$ $\forall f,g\in X=\{h:\mathbb{R}\rightarrow\mathbb{C}:$ $h$ is Lebesgue-measurable and bounded over $\mathbb{R}$} I have to ...
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37 views

Show that a Banach space $E$ is reflexive $\iff$ for all closed subspace $F$ of $E$ we have $F^{\perp \perp} = J(F)$

Problem: Let $E$ be a normed space over $\mathbb{C}$. Show that we are able to embed $E$ into the second dual space (bidual) $E''$ of $E$ by a linear isometry i.e we can consider $E$ as a subspace of ...
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compute the norm of a linear operator

good morning everyone, I am preparing my functional analysis exam and I can't resolve this exercise: Let $$ H = L^2([0, 3])$$ and let T : H → H be defined by $$Tu(t) = (1 + t^2)u(t)$$ i) Prove ...
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25 views

Practical approach/example for a paper [closed]

My problem is, that I cant say what the different definitions in the paper means in my practical example. My example, and I think its the easiest one, is the process of making a photo, save the ...
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29 views

Linear Operator to vector corrrespondanc through change of basis

Consider a linear map as follows $$vec: L(X,Y): Y \otimes X $$ $$ vec: E_{b,a} \mapsto e_b \otimes e_{a}$$ which can be looked as a change of a basis map. Where $E_{b,a}$ is usual basis for $L(X,Y)$ ...
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whether span of orthonomal basis equals Hilbert space

For example, there is a Hilbert space H. And {$e_{i}$} is the basis of this space. We know that every f $\in$ H can be represented by linear combination of {$e_{i}$}. So why do we use $\overline{...
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Is $\left\{ f(x), e^{2\pi i x }f(x) \right\}$ is Linearly independent in $L^2(\mathbb R)$? [closed]

Let $0\neq f \in L^2(\mathbb R)$ (complex Hilbert space). (1) Can we say that $\left\{ f(x), e^{2\pi i x }f(x) \right\}$ is Linearly independent in $L^2(\mathbb R)$? (2) Is there any known ...
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About Riesz representation theorem

Let $X=l_p^{(3)}$, where $1\lt p \lt \infty$, and $\phi(x) = x_1-2x_2+3x_3$. Decide whether $\phi$ is bounded, and if so, find $||\phi||$. So by marking $y=(1,-2,3)$, we can see that $\phi(x)=\...
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What is the definition of a bounded operator in an infinite dimensional Hilbert Space?

I am struggling to understand the meaning of a bounded operator in a Hilbert Space. Does a bounded operator simply means that if it acts on an element of the Hilbert Space, the "result" is bounded?
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Show that for $|a|<1$ the series $\sum_{n=1}^{\infty}a^nf_n$ converges in $L_2([-1,1])$.

Let $f_n(x)=x^{n+1/2}$ for $n\geq 1$ and $x\in [-1,1]$. Show that for $|a|<1$ the series $\sum_{n=1}^{\infty}a^nf_n$ converges in $L_2([-1,1])$. First, I have shown that $\sum_{n=1}^{\infty}a^nf_n(...
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Isometric isomorphism between $L^2$ and $\mathcal{L}^2$

I was reading and trying to understand the proof that the space $\mathcal{L}^2 (\mathcal{H})$ (Hilbert-Schmidt operators) is made by all the $T_K:L^2(X,\mu) \rightarrow L^2(X,\mu)$ with $K \in L^2(X \...
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Spectral Theorem for Unitary Operator

It is well known that the following - in many literature - called the Spectral Theorem for Unitary Operator. I would like to know where i can find further information about it and its proof.
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Spectrum of operator A

Let $H$ be a Hilbert space and $A \in B(H)$. How to prove that if $|\lambda| = \|A\|$ and $\lambda \in W(A)$, then $\lambda \in \sigma_p(A)$? Here $W(A)$ is the numerical range of the operator $A$: $$...
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Proof that any sequence in $\mathbb{C}^k$ converges iff each component sequence converges.

I have attempted to prove that every sequence in $\mathbb{C}^k$, equipped with the standard inner product, is convergent if and only if each component sequence is convergent. I am using this result as ...
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$H$ Hilbert space, $T$ symmetric bounded linear, when is $H=R(T) \oplus N(T)$?

I just saw in an exercise that if I have a prehilbert space $H$ and $T$ a linear, bound and symmetric operator then $R(T)=N(T)^{\perp}$. Now I was asking myself whether $H=R(T) \oplus N(T)$. On wiki I ...
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$F(L_2([0,1],\mu))$ is dense in $L_2([0,1],\mu)$, where $F$ is the multiplication operator

Consider $A=L_2([0,1],\mu)$ where $\mu$ is the Lebesgue measure. Let $F:A\rightarrow A$ be linear given by $Ff(x)=xf(x)$ for $f\in A$ and $x\in[0,1]$. I want to show that $F(A)$ (the range of $F$) is ...
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analysis of $T : f \to Tf$ with $[T(f)](x) = ie^{i\pi x}(\int_0^x e^{-i\pi t}f(t)dt - \int_x^1 e^{-i\pi t}f(t)dt)$

$\, f \in L^2([0,1],\mathbb{C})$ show that $T : f \to Tf, \, f \in L^2([0,1],\mathbb{C})$ is continuous, $[T(f)](x) = ie^{i\pi x}(\int_0^x e^{-i\pi t}f(t)dt - \int_x^1 e^{-i\pi t}f(t)dt)$ the ...
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Prove that $(Au)(t)=\frac{d^{2}u(t)}{dt^{2}}$ is self-adjoint

Let $D(A)=\{ u \in L_2(0,T)| u, \frac{du}{dt}$ are absolutly continuous with $\frac{du}{dt} \in L_2(0,T)$, $u(0)=u(T)=0\}$ and, $(Au)(t)=\frac{d^{2}u}{dt^{2}}$ prove that $A$ is self-adjoint. Trial ...
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show that $\ker(Id-T) = \ker(Id-T)^{\star}$

$H$ is a hilbert space and $T$ is a bounded linear operator on $H$, also $\|T\| \leq 1$ by calculating $\|Tx-x\|^2$ I have shown the following string of equivalences $$Tx = x \iff \langle\,Tx, x\...
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Real positive index Sobolev spaces are Hilbert spaces

I'm trying to prove that, for $k\geq 0$, Sobolev spaces defined in this way: $H^k(\mathbb{T})=\{f\in L^2(\mathbb{T}): \sum_{n=-\infty}^{+\infty}(1+n^2)^k|\hat{f}(n)|^2 < +\infty\}$ are Hilbert ...
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What is the operator norm of $Tf(x) = x^2f(x)$?

Let $H = L^2([0,1],\mathbb{R})$ and $T : H \to H,\, Tf(x) = x^2f(x) $. $T$ is linear. $$\|Tf\|_{L^2([0,1],\mathbb{R})} = \sqrt{\int_0^1x^4f^2(x)dx} \leq\sqrt{\int_0^1f^2(x)dx} = \|f\|_{L^2([0,1],\...
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Show an inequality

Show that on the Hilbert space: $$x\bot y $$ if and only if $$ \Vert x + Cy\Vert \ge \Vert x \Vert,$$ $\forall C\in \mathbb R.$
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Reference for spectral theorem for normal operators.

Does anyone know of a good reference for the spectral theorem (projection valued measure version) for possibly unbounded normal operators? I would also be interested in examples where this sort of ...
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26 views

Boundedness and Strong convergence

$f_n\rightarrow f$ in $L^2(0,1)$, $\{ f,f_1,f_2,\ldots \}\subset H^1(0,1)$, $||f_n||_{H^1(0,1)}\leq M,\ \forall n\geq 1 $, Is is true that $f_n\rightharpoonup f$ in $H^1(0,1)$? If not, then what is a ...
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33 views

Prove $Tx=x$, for $x\in H$, if and only if $(Tx,x)=\|x\|^2$ and $\ker(I-T)=\ker(I-T^*)$

Let $H$ be a complex Hilbert space and $T:H\rightarrow H$ an operation such that $\|T\|\leq 1$. Show that $Tx=x$ if and only if $(Tx,x)=\|x\|^2$ $\ker(I-T)=\ker(I-T^*)$. My attempt 1. ...
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1answer
41 views

Mercer's Theorem importance (Kernels)

I understood that Mercer's Theorem extends the definition of kernels also for infinite input space. In Machine Learning realm our training set is always finite and hence the input space is always ...
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48 views

Weak convergence of bounded sequence $(x_n)$ in Hilbert space where $\langle{x_n,y\rangle}\rightarrow \langle{x_n,y\rangle}$ for all $y\in D\subset H$

Let $H$ be a Hilbert Space endowed with the inner product $\langle{.,.\rangle}$ and $D$ a subset of $H$ such that span$(D)$ is dense in $H$. Show that, given a bounded sequence $(x_n)$ in $H$, such ...
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28 views

A complete orthonormal system $\{e_i\}^\infty_{i=1}$ in $H$ is a basis in $H$

I'm studying about Hilbert space from a book of functional analysis and I just read this theorem (2.1.10) and its' proof. I cannot understand why $(y-x)\perp e_i$? why is it implied?