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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Proof: $C^\infty[0,1]$ is dense in $L^2[0,1]$

Intro: I would like to know if my demonstration of $C^\infty[0;1]$ is dense in $L^2[0,1]$ is correct because I didn't find any complete demonstration of that statement. -(i) As we know from here all ...
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Why is the following not a Hilbert basis

Why is the set {($z^n$ → $\frac{z^n}{||z^n||}$)} not a Hilbert Basis on $L^2$ ($\mathbb{D}$,dz) where dz is the Lebesgue measure and D := {(z $\in$ $\mathbb{C}$| |z|< 1}. I have no idea on how to ...
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3 votes
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Proof: All functions in $L^2[0,1]$ are in $L^1[0,1]$

I would like to know if my demonstration of all functions in $L^2[0,1]$ are in $L^1[0,1]$? $\forall f\in L^2[0,1] $ we can split $f$ in two differents spaces: $A=\left \{0\leq x\leq 1:|f(x)|>1) \...
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If $A$ is a generator of a strongly continuous semigroup and $B$ a bounded linear operator then $A+B$ generate a strongly continuous ssemigroup?

Let $X$ Banach and $A \colon D(A) \subset X \to X$ be a generator of a strongly continuous semigroup $e^{At}$. Let $B \in L(X)$ (linear bounded opearator on $X$) we then know that $B$ generates a ...
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Aren't all eigenvalues of $T^*T$ non-negative? (Clarification regarding the definition of singular values)

I'm having some confusion with the definition of the singular value of a matrix. As per Wikipedia: In mathematics, in particular functional analysis, the singular values, or s-numbers of a compact ...
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Galerkin Methods

Let $(\varphi _n)_{n\in \mathbb{N}}$ be a basis of an Hilbert space $H$. Define $H_N=\operatorname{span}\{\varphi _1,\ldots ,\varphi _N\}$. Suppose that there exits $a:H\times H \to \mathbb{R}$, a ...
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$x=Px+Qx$ unique decomposition in Hilbert space

Problem: Let $M$ be a closed subspace of Hilbert space $H$. Then every $x \in H$ has unique decomposition $x = Px + Qx$ where $Px \in M$, $Qx \in M^{\perp}$ . I don't understand: When proving that our ...
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Zero condition for bounded operators on Hilbert space

I found in a math textbook this exercise: show that, if A is a linear bounded operator on a complex hilbert space H and <x|Ax> = $0$ for every x in H, then A= $0$. Why the operator is required ...
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Why is a compact linear operator also a bounded linear operator?

I know that a linear operator is bounded iff continuous, with these definitions of "bounded" and "continuous": a linear operator T (on H Hilbert space) is bounded if ∃M>0: ||Tf|...
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Doubt from the proof that the sequence space $l^2$ is a Hilbert space from Kreszig book

This is the proof given in Kreszig book. 3.1-6 Hilbert sequence space $l^2$. The space $l^2$ (cf. 2.2-3) is a Hilbert space with inner product defined by $$ \langle x, y\rangle=\sum_{j=1}^\infty \...
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2 votes
1 answer
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When is the space of linear and bounded operators between Hilbert spaces a HIlbert space?

Let $H_{1}$ and $H_{2}$ be two Hilbert spaces. My question is, when is the set of linear and bounded operators $\mathcal{L}(H_{1}, H_{2})$ with the usual norm a Hilbert space? I think I've proved that ...
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Invertible linear operators without eigenvalues [duplicate]

Suppose $V$ is a vector space over the complex numbers, and let $\alpha: V \mapsto V$ be an invertible linear operator. If the dimension of $V$ is finite, we know that $\alpha$ has (nonzero) ...
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Question regarding conjugate operators and the harmonic operator.

Let us consider the operator $\hat{n}=\hat{a}^\dagger\hat{a}$ as the number operator of the harmonic oscillator. Let $|n\rangle$ be the eigenstates. Then we can say : $$\hat{n}|n\rangle=n|n\rangle$$ I'...
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Question about direct sum in a Hilbert Space

What is the intuition for looking at the vector $h^\star - th$? I understand we choose an arbitrary element $h\in V$ to show that some vector is orthogonal to every element of $V$, but why might we ...
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Prove that if $TM_{e^{2\pi ix}}=M_{e^{2\pi ix}}T$ then $f=0$ where $T=C_f$ (convolution), $f \in C[0,1]$ and $M_g(h) := gh$

I am trying to solve the following problem: Let $H=L^2[0,1]$, $T=C_f$ (convolution) where $f \in C[0,1]$ (continuous function). Define $M_g(h) = gh$. Prove that if $TM_{e^{2\pi ix}}=M_{e^{2\pi ix}}T$ ...
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Eigenvalues of compact self adjoint operator that is cyclic

Let $T: H \to H$ be an arbitrary compact self adjoint operator, and $H$ a Hilbert space. I am asked to prove that $T$ being cyclic is equivalent to all of the eigenvalues of $T$ having multiplicity ...
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1 vote
1 answer
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On the existence of the Moore-Penrose inverse

The following was written in a paper, but I couldn't find out why. Does anyone have any idea on how to prove this claim? It is well known that $A^{\dagger}$ exists for a given $A \in B(H, K)$ if and ...
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2 votes
1 answer
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Using the completion of $ T:PC[0,1]\to PC[0,1]$ to $ \hat{T}:L^2[0,1]\to L^2[0,1]$ in order to find the image of $\hat{T}$

Given a bounded linear operator: $$ T:PC[0,1]\to PC[0,1]$$ and its completion $$ \hat{T}:L^2[0,1] \to L^2[0,1]$$ (defined by $\hat{T}x:=\lim Tx_n$ where $x_n \to x$). If $T$'s image is $C[0,1]$ (all ...
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Existence of system of PDE using Lax Milgram

I am trying to apply the Lax Milgram theorem to a System of PDE. Therfore I would like to apply it on the cartesian Product $H_{0}^{1}\times H_{0}^{1}$. This is a Hilbert space with the Norm $\sqrt{\|\...
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Is the orthogonal complement projector a projection operator?

Let $X$ Hilbert space and $U \subset X$ convex closed. Then we can define the projection operator on $U$ $P \colon X \to X$ such that $P x \in U$ is the orthogonal projection of $x \in X$. Moreover $P$...
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Function in $H^1(\mathbb{R})$

Consider the function $$f=|x|^pe^{-x^2},$$ where $p$ is a real constant. The function $f$ is in $L^2(\mathbb{R})$ iff $p>-1/2$. The function $f$ is in $H^1(\mathbb{R})$ iff $p>1/2$ or $p=0$. I ...
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2 answers
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If $\lambda$ is in the continous spectrum of $L$, then the range of $L-\lambda$ can not be closed?

Let the continuous spectrum of a densely defined linear operator $L$ over a Separable Hilbert space, be defined as the set of all $\lambda \in \mathbb C$ such that: (i) $L-\lambda$ is injective, (ii) ...
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Weak convergence in Hilbert space, convergence in distribution and pointwise convergence

I am learning about weak convergence in Hilbert spaces of functions and I am wondering if it is the same as pointwise convergence or convergence in distribution. I do not really see a difference ...
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Bounded inverse for a closed range closed operator

I am dealing with an unbounded operator $T$ on an Hilbert space $H$. I am interested in proving that it has a bounded inverse $T^{-1}$. I managed to prove that the operator is closed and self-adjoint. ...
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Uniqueness of a projection onto a subspace in Hilbert space

I need to show that $P_{K}(v) = argmin_{w \in K} ||v-w|| $ (projection) exists and is unique given $K$ is a closed convex subset of a Hilbert space $(V,<.,.>)$ and $v \in V$. I know the ...
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T:H-->F is a bounded linear functional..where F=R(real) or C(complex) Now,||T(x)||=||T||_op (operator norm) this is it be true that ||x||>1??

T:H-->F is a bounded linear functional..where F=R(real) or C(complex) Now,||T(x)||=||T||_op (operator norm) now what can be told about ||x||?? My guess it should be ||x||<=1.but also i can't ...
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Gradient of a functional defined on an Hilbert space (with respect to a $W^{1,2}$ inner product)

$\newcommand{\R}{\mathbb R}$ Consider the Hilbert space $X = W^{1,2}(\R)\oplus W^{1,2}(\R)$ (Sobolev spaces). I define a function $F:X\to \R$ as $$F(u,g)= \int_\R u(t)\partial_tg(t) dt.$$ $F$ is ...
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5 votes
2 answers
160 views

Definition of Schauder basis

I have a definition of a Schauder basis but I’m unsure of it. The definition I have is A sequence $\{e_k : k \in \mathbb{N} \}$ in a normed space $(V, \| \cdot \| )$ is a Schauder basis if $\sum_{k=1}...
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2 votes
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Orthogonal Complement of the $\pi(H)$-invariant Vectors in a Hilbert Space

Let $H \le G$ be an inclusion of countable, discrete groups, and let $\pi$ be unitary representation of $G$ on some Hilbert space $\mathcal{K}$. Is it possible to determine the orthogonal complete $(\...
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Projection mapping as quadratic programming problem.

I am studying Variational inequalities in Hilbert space. To define this, let $C$ be a closed, convex subset of a Hilbert space $H$. The Variational inequality problem is to find $x \in C$ such that $\...
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Proof of minimum value not equal to zero in linear bounded operator mapping(Hilbert space)

I am new to operators and Hilbert spaces in functional analysis. I am trying very hard to prove the below question but could not do it. It is an exercise problem in my university course. Could anyone ...
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1 vote
0 answers
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Functional analysis - best approximation norm [closed]

I have no idea how to prove or disprove the parts of the below question. I intuitively feel they are correct but cannot form a rigorous mathematical proof for it. Given is a normed space E. Also, ...
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2 votes
1 answer
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Equivalence between two definitions of hermitian adjoint

Given the two definitions of hermitian adjoint: $(1): <\Psi|A|\Phi>=(<\Psi|A^+|\Phi>)^*$ $(2): <\Psi|A\Phi>=<A^+\Psi|\Phi>$ I want to show that they are equivalent However I ...
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Projective Limits of Hilbert Spaces

Context: Suppose that we have an inverse system $(H_n, \varphi_{m,n}: H_n \to H_m)_{m \leq n}$ of Hilbert spaces indexed by $\mathbb{N}$, where for now we just assume that the $\varphi_{m,n}$ are ...
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1 answer
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Orthogonal complement of kernel of linear functional has dimension n iff T surjective

I have successfully proven that for every linear bounded map $T:H \to \mathbb{K}^n$ it holds that $\dim(\ker(T)^\bot)\leq n$ where $H$ is a hilbert space. Now, I need to prove that equality holds iff $...
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4 votes
0 answers
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Derivative of a map $f:\mathbb R \to \ell^2 $ to a separable Hilbert space vs derivative of each component of the Hilbert basis.

${\newcommand{\R}{\mathbb{R}}}$ Let $\ell^2 $ be the Hilbert space of square summable sequences of real numbers. Consider a map $f: \R \to \ell^2$, that has components $f_n:\R\to \R$, i.e. $f(t)=\{...
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0 answers
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Why does this "proof" using Gaussian Elimination on an infinite sequence of vectors not work?

Consider the following problem from this (very awesome!) video. If $\alpha \in \mathbb{C}$ has $0 < |\alpha| < 1$, and defining the vectors $f_k = (1, \alpha^k, \alpha^{2k}, \alpha^{3k}, \dots) \...
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0 answers
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Finding a Kernel Matrix on $ \mathbb{R}^n$

Let $ X = \mathbb{R}^n$ with inner product $⟨x, y⟩ = y^TQx$ for some symmetric positive definite matrix $Q$. I have a few questions about this setup and generally about the kernels. 1- A kernel is the ...
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3 votes
2 answers
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An identity of inner product

I am trying to prove the following identity $$\|\psi\|^2=\|A(A+iaI)^{-1}\psi\|^2+a^2\|(A+iaI)^{-1}\psi\|^2$$ where $A$ is self-adjoint and $a$ is a real number. My approach is to rewrite $A(A+iaI)^{-1}...
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2 votes
2 answers
47 views

Bounded operators: $\langle T_n (x),x \rangle \longrightarrow 0 \implies T_n (x) \rightarrow 0$?

Suppose $(T_n)_{n=1}^\infty$ is a sequence of bounded operators on a Hilbert space $\mathcal{H}$ and that for every $x \in \mathcal{H}$ $$ \langle T_n (x),x \rangle \longrightarrow 0 \quad \text{as } ...
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1 answer
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Does equality in distribution of stochastic fields at each point, imply equality in distribution of integral over the field of both the random fields?

Let $(D,\mathcal{B}(D),\sigma)$ be some measurable space, where $D\subset R^n$, $\mathcal{B}(D)$ is the borel sigma algebra, and $\sigma$ is the lebesgue measure. Let $T_1,T_2\in L^2(\Omega;L^2(D))$. ...
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2 votes
1 answer
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If inequality holds in $L^2(E)$, then it must hold in $L^2(F)$ when $E \subseteq F$.

Suppose that $E \subseteq F$ are measurable and bounded sets. Suppose that for every finite sequence $c$ we have $$A \sum_n |c_n|^2 \leq \left\|\sum_n c_n e^{2 \pi i \alpha_n x}\right\|_{L^2(E)}^2 \...
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0 votes
0 answers
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How does $V \geqslant I$ imply that $V$ is invertible?

I am reading the proof of Hildebrandt's theorem in these notes by Paul Skoufranis. The proof of Proposition $4.1$ (Rota's theorem) on Pg. $13$ says that: Moreover, we notice that $T^*VT = V - I$. ...
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Difference in two proofs of a claim involving orthogonal systems

Apologies for the vague title, but there's quite a bit of information in this question and I wasn't sure how to condense it down into a single title. In this question, we wish to devise a proof for ...
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1 vote
0 answers
31 views

How to prove the space $L^2\big([a,b],w(t)\big)$ is complete and separable?

Suppose $w(t)$ is a positive and measurable function on $[a,b]$. If $x(t)$ is a measurable real function on $[a,b]$ satisfying $$ \left\| x \right\|^2 =\int\limits_{\left[ a,b \right]}{w\left( t \...
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0 answers
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Identify a separable Hilbert space with the space $\mathcal l^2$

Let $X$ separable Hilbert space with orthonormal basis $(e_i)$. Then any $x \in X$ can be written as $$x=\sum_{\mathbb N}x_i e_i$$ for $x_i \in \mathbb R$ and thus it is quite natural to identify $X$ ...
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0 votes
1 answer
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How to prove that the range of this integral operator $T$ is $L^2[a,b]$?

The operator is $Tf= g(t)+ \int_{a}^{t}(K(t,s)f(s)ds.$ The proof assumes that $g(t) \in L^2[a,b]$ so I believe I need to only prove that $\int_{a}^{t}(K(t,s)f(s)ds \in L^2[a,b]$. I started off this ...
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1 vote
1 answer
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The space of Hilbert-Schmidt operators form a Banach space

I am having trouble proving the following result: Show that the space $X$ of bounded operators on a separable Hilbert space into itself for which the Hilbert-Schmidt norm is finite, is a Banach space ...
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1 vote
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Exercise 5.16 from Brezis' Functional Analysis

Suppose $H$ is a Hilbert space with scalar product $(\cdot , \cdot )$ and norm $| \cdot |$. Let $V \subseteq H$ be a dense linear subspace and say it has its own norm $\| \cdot \|$ such that $V$ is ...
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2 votes
0 answers
37 views

Example of the continuity of the Fredholm index

Consider the space of square-summable sequences $\ell^2 =\{(a_0,a_1,\ldots) : \sum_{n=0}^\infty |a_n|^2<\infty\}$. Let $I$ denote the identity on $\ell^2$ and define the unilateral shift operator \...
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