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
162 views

Is every symmetric bilinear form on a Hilbert space a weighted inner product?

Is every symmetric bilinear form on a Hilbert space a weighted inner product? i.e. can I write that $b(u,v) = (wu,v)_H$ for all $u, v \in H$? I am not sure about this. Maybe something to do with Riesz ...
2
votes
1answer
375 views

self-adjoint operator proof

Let $T$ be a densely defined closed unbounded operator on a Hilbert space $H$. A number $\lambda \in C$ is called an approximate eigenvalue of T if there is a sequence ${X_n} \subset D(T)$, with ...
2
votes
1answer
163 views

Orthogonal projections for minimization problem

I have trouble to understand the existence of a minimization problem in a Hilbert space. Let $(\Omega,\mathcal{F}_T,P)$ be a filtred probability space with filtration $(\mathcal{F}_t),0\le t\le T$. We ...
4
votes
1answer
186 views

Spectrum proofs

Let $T$ be a densely defined closed unbounded operator on a Hilbert space $H$. Show that if $\lambda$ is a point in the residual spectrum of $T$, then $\bar{\lambda}$ is in the point spectrum of the ...
0
votes
1answer
92 views

proof related to Hilbert Spaces

Let $T$ be a bounded linear compact operator on a Hilbert space $H$ over $C$, $A$ is a positive self-adjoint operator on $H$. How to show that $T=UA$ where $U^{+}U=I$ on the range $R(A)$ of $A$
2
votes
1answer
93 views

Limit of a sequence in the space $\ell_2$

I have difficulties in the following problem. Let $H=\ell_2$ be the space of square-summable sequences. Let $\alpha\in (0,1)$ and $\{u^k\}\subset H$ be such that $$ u^{k+1}=(1-\alpha)u^k+\alpha ...
1
vote
1answer
107 views

exercise: limit orthonormal sequence, “Banach Space Theory”

I have an exercise from "Banach Space Theory": Suppose $\{x^k\}_{k=1}^\infty$ is an orthonormal sequence in $l_2$, where $x^k:=(x_i^k)$. Show that $\lim_{k\rightarrow \infty} x_i^k =0 \ \forall_{i\in ...
2
votes
1answer
147 views

Computing an explicit solution to an integral equation via the Neumann Series.

I am hoping that someone can provide guidance for solving the integral equation $$u=f+\lambda Au$$ where $1/\lambda\notin\sigma(A)$, $f\in L^2[0,2\pi]$, and $A:L^2[0,2\pi]\to L^2[0,2\pi]$ is defined ...
2
votes
0answers
55 views

Prove that for every $f$ in $H$, the sequence $u_n$ which is the projection of $f$ on $K_n$ converges to a limit

$K_n$ is non-increasing sequence of closed convex sets in Hilbert space $H$ such that the intersection of $K_n$ is different from emptiness. Prove that for every $f \in H$, the sequence $u_n$ which is ...
1
vote
1answer
152 views

Self-adjoint operator on a Hilbert space.

Let $T$ be a self-adjoint operator on a Hilbert space $H$. If for all $x\in H$, $\langle Tx,x\rangle=0$, is $T=0$?
0
votes
1answer
406 views

Is my proof of characterisation of self-adjoint operators on complex Hilbert spaces okay?

I wish to show the following theorem: Let $T:H\to H$ be a bounded linear operator on a complex Hilbert space $H$. Then if $\left\langle Tx,x\right\rangle \in\mathbb{R}$ for all $x\in H$, then $T$ is ...
1
vote
1answer
66 views

Orthogonal Projectors

Please, I need help with this proble. Let $(H,\langle\cdot,\cdot\rangle)$ be a Hilbert space and let $V_1,V_2,\ldots,V_N$ closed subspaces, mutually orthogonal of $H$, that is, $v_i\perp v_j$ ...
1
vote
2answers
161 views

Hahn-Banach theorem (second geometric form) exercise #2

Let $X$ be a Hilbert space and $\{F,F_1,\ldots,F_N\}$ linear functionals over $X$ such that $$\bigcap_{i=1}^N\mbox{ker}(F_i)\ \subseteq \mbox{ker}(F),$$ and any kernel of the involved functionals is ...
2
votes
1answer
44 views

Two isomorphic inner product spaces, one is complete, is the other also complete?

If you two have inner product spaces and one is complete, and there is an isomorphism between the two spaces, is the other space also complete? Or do we absolutely require equivalence of norms?
3
votes
1answer
190 views

Hahn-Banach theorem (second geometric form) exercise

Let $X$ be a vector normed space and $\{F,F_1,\ldots,F_N\}$ linear functionals over $X$ such that $$\bigcap_{i=1}^N\mbox{ker}(F_i)\ \subseteq \mbox{ker}(F).$$ Apply the Hahn-Banach theorem (second ...
1
vote
1answer
140 views

Riesz representation theorem on Hilbert space with equivalent norms

If we have a Hilbert space that has two equivalent norms (and inner products), are the Riesz maps (from Riesz representation theorem) associated with each inner product the same?
0
votes
1answer
59 views

Equivalent norms and density/separability

$V \subset H$ are Hilbert spaces with inner products $(\cdot,\cdot)_V$ and $(\cdot,\cdot)_H$. Suppose $V$ is dense in $H$ and both spaces are separable. If $(\cdot,\cdot)_{V_2}$ and ...
1
vote
2answers
96 views

Sequence of operators in a Hilbert space

The question is: Let $H$ be a Hilbert space and $\{T_n\}$ be a sequence in $B(H)$ such that $\lim_{n\rightarrow\infty}\langle x, T_n y \rangle = 0$ for all $x, y \in H$. Prove or disprove $\sup_n ...
2
votes
2answers
74 views

little question about linear operators

Let H be a complex Hilbert Space. Let $P \in L(H)$ be an idempotent operator ($P^{2} = P$). Also, let $\parallel P\parallel = 1$. I want to prove that $P$ is an orthogonal operator. I defined $M = ...
2
votes
1answer
449 views

How can I able to show that $(S ^{\perp})^{\perp}$ is a finite dimensional vector space.

Let $H$ be a Hilbert space and $S\subseteq H$ be a finite subset. How can I able to show that $(S ^{\perp})^{\perp}$ is a finite dimensional vector space.
1
vote
1answer
93 views

Orthogonal family in Hilbert Space

Let $(x_k)_1^\infty$ be an orthogonal family of points in X a Hilbert space. Then $\sum_{i=1}^\infty x_i$ converges if and only if $\sum_{i=1}^\infty ||x_k||^2$ converges. Also need to show that ...
1
vote
1answer
56 views

some inclusions regarding linear operators

Let $H$ be a Hilbert Space and $T:H\rightarrow H$ a linear operator. Let $T^*$ be the adjoint operator of $T$ and let $\operatorname{Cl}(X)$ be the topological closure of the set X and $X^{\perp}$ ...
0
votes
1answer
29 views

Does this dual space functional pairing = 0 imply functional = 0?

If $V$ is a Hilbert space, is it true that if $\phi_1, \phi_2 \in C_c^\infty(0,T)$, $$\int_0^T \langle \phi_1(t)g +\phi_2(t) f, v \rangle_{V', V} = 0$$ for all $v \in V$, then $\phi_1g + \phi_2f ...
1
vote
0answers
35 views

Weighted inner product space and representation of dual space

Let $H$ be a Hilbert space and define $H_c$ to be the weighted Hilbert space with inner product $$(u,v)_{H_c} = c(u,v)_H$$ where $c$ is a positive constant. Then is it true that $$c\langle f, u ...
1
vote
0answers
105 views

Countable orthonormal basis of product of separable Hilbert spaces

If I have 2 separable Hilbert spaces $X$ and $Y$ which have (different) orthonormal bases $x_i$ and $y_i$, then clearly $x_i \times y_j$ is a basis for $X \times Y$ (which is also a separable space). ...
1
vote
1answer
237 views

Riesz Representation theorem-pde

Consider $\sum_{i,j=1}^n \displaystyle\int_{\mathbb{R}^n} \dfrac{\partial^2 u}{\partial^2 x_i} \overline{\dfrac{\partial^2 v}{\partial^2 x_j} } dx + \lambda \displaystyle\int_{\mathbb{R}^n} u ...
7
votes
1answer
320 views

Approximating a Hilbert-Schmidt operator

Let $H$ be a separable Hilbert space. Recall that a bounded operator $A : H \to H$ is said to be Hilbert-Schmidt if $$\|A\|_{HS}^2 := \sum_{i=1}^\infty \|A e_i\|^2 < \infty$$ where ...
1
vote
1answer
69 views

Why is $\langle f, u \rangle_{H^{-1}, H^1} = (f,u)_{L^2}$ when $f\in L^2 \cap H^1$ and not $\langle f, u \rangle_{H^{-1}, H^1}=(f,u)_{H^1}$?

More generally, if $V \subset H \subset V'$ are Hilbert spaces, why is $$\langle f, u \rangle_{V',V} = (f,u)_{H}$$ when $f\in H \cap V$ and not $$\langle f, u \rangle_{V',V}=(f,u)_{V}?$$ Is this what ...
2
votes
2answers
90 views

True or False; Functional Analysis

Given $T: V \to W$ with $V,W$ being Hilbert Spaces. We always have $\| T^ *\| = \| T \|$. I think it is true because of Riesz' Theorem, but I am not sure if a proof is necessary. EDIT: In case ...
1
vote
0answers
43 views

Weak limits and structure of a generated semigroup

I am getting acquainted with the beautiful theorem known as Jacobs–de Leeuw–Glicksberg decomposition. A special case of this theorem is the following: Theorem. (Jacobs–Glicksberg–de Leeuw ...
1
vote
1answer
180 views

Bounded linear operator in weak topology

Let $B$ be a bounded linear operator on $H$. Prove $B\colon (H,w)\to (H,w)$ is continuous. $(H,w)$ is a Hilbert space with its weak topology.
3
votes
0answers
200 views

Prove this equality in a Hilbert Space $H$ with Riesz's Representation Theorem.

For $x \in H$, prove that $\sup_{\|z\| = 1} \langle x , z \rangle= \| x \|$ Here is a quick one. If someone could improve this it would be great Proof By Cauchy Schwarz, $\langle x,z \rangle ...
1
vote
1answer
119 views

Is the Strong Limit of a Linear Operator in a Hilbert Space the Same as the Norm Limit?

If $H$ is a Hilbert Space, and I have an operator $F:H \rightarrow H$ which is the limit of a sequence of operators $F_n$ with respect to the operator norm; and this same sequence of operators ...
1
vote
2answers
264 views

Weak convergence-exercice

Let $\Omega$ be an open set in $\mathbb{R}^n$ and let $(u_n)$ be a bounded sequence in $H^1_0(\Omega).$ Who's the theorem say that we can extract a subsequence denoted $u_{n}$ as $u_n$ weakly ...
0
votes
1answer
118 views

Dual space of product of Hilbert spaces

What can we say about the dual space of the product of Hilbert spaces? Suppose $V = A \times B$ and the inner product of two vectors in this space is the obvious one (add the two separate inner ...
0
votes
1answer
44 views

Functional on Triangulation

Let $\Omega$ a bounded open subset of $\mathbb{R}^2$ and let $\mathcal{T} :=\{T_1,T_2,\ldots,T_N\}$ a triangulation of $\Omega$, i.e, $\overline{T}_i$ is a triangle with non empty inner, $\forall ...
0
votes
1answer
51 views

What is bounded point evaluation property?

I read an article, it contains sentence like this - The hilbert space A possesses the bounded point evaluation property. What does this mean? I found this Meaning of Point Evaluation, is it connected ...
1
vote
1answer
93 views

Study the equivalence of these norms

I have two Hilbert spaces $H_1$ and $H_2$ and I consider a set of functions $f$ which decompose as $f=g+h$ with $g\in H_1$ and $h\in H_2$. I know that this decomposition is unique. So I define the ...
6
votes
1answer
211 views

Matrix Representation of Trace Class Operators

Suppose we have a separable Hilbert space (thus with a countable basis) and that represent an operator in matrix form, i.e: $A: H \rightarrow H $$$x \;\rightarrow \sum_{j \in \mathbb{N}}\left(\sum_{k ...
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vote
0answers
148 views

How can projection operators be limits of powers of unitary operators?

Consider a (fixed) unitary operator $U$ acting on the Hilbert space $\mathcal{H}$. Because the unit ball is compact in the weak topology, it is not hard to see that there exists a (smallest) compact ...
2
votes
1answer
133 views

Show for compact operator $K$, if $||Kf|| < ||f|| \forall f$, then $||K|| < 1$.

I wanted to check my reasoning on proving this statement, and see if anyone had suggestions for other proofs of this fact. Again, the statement is, if $K$ is a compact operator on a Hilbert space ...
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vote
0answers
64 views

Verify solution: Is this gradient, correct?

For a function $$f(X)=\operatorname{tr}(X^TAX)+\|\operatorname{diag}(X^TX)-\alpha I\|_2,$$ where all entries are real and $\alpha$ is a real scalar, while $A$ is a p.s.d matrix and $X$ is a real ...
1
vote
0answers
103 views

Geodesic on a Hilbert manifold

Given a Hilbert manifold $\mathcal H$ (always using the natural Hilbert inner product) and a geodesic $\Gamma(t)$ in this manifold, can one show that the projection of this geodesic onto a submanifold ...
1
vote
1answer
50 views

Is this function in the space $L^1$?

I have this function $$f(x)=\frac{1}{\vert x-y\vert^2(1+\vert x\vert^2)^s}$$ with $x\in\mathbb{R}^3$ and $y$ a fixed point. I have to study for which values of $s>0$ it belongs to ...
3
votes
0answers
235 views

Prove Heisenberg uncertainty principle (measure and integration theory)

Here is a question in measure and integration theory, Let $f$ be a continuously differentiable complex function on $\mathbf{R}$ s.t. the functions $x \mapsto xf(x)$ and $f'$ are in ...
2
votes
1answer
39 views

Quick question about sum of subspaces of a Hilbert space

I just have a quick question. Suppose there is $Z$, a Hilbert space, with $A$ and $B$ closed linear subspaces. If $(a,b)=0$ for all $a \in A$ and $b \in B$, I know that $A+B$ is also closed. I don't ...
1
vote
1answer
585 views

Calculating the Norm of an operator in $L^2(0,1)$

If I have the following operator for $H=L^2(0,1)$: $$Tf(s)=\int_0^1 (5s^2t^2+2)(f(t))dt$$ and I wish to calculate $||T||$, how do I go about doing this: I know that in $L^2(0,1)$ we have that ...
0
votes
0answers
115 views

Showing an operator is self adjont

I am trying to show that the operator: $$Tf(s)=5s^2\int_0^1t^2f(t)dt+2\int_0^1f(t)dt$$ is self adjoint where $H=L(0,1)$ with real scalars and $t\in \mathcal{L}(H)$. So I can re-write this operator ...
8
votes
1answer
135 views

Is $\mathcal{C}([0,1])$ homeomorphic to a Hilbert space?

Let $\mathcal{C}([0,1])$ the Banach space of continuous functions from $[0,1]$ to $\mathbb{C}$. The norm on $\mathcal{C}([0,1])$ is $f \mapsto \| f\|_{\infty}= \sup_{x \in [0,1]} |f(x)|$. Is it ...
2
votes
1answer
101 views

Is my proof correct? I want to show if $V \subset H$ is dense, then $L^2(0,T;V) \subset L^2(0,T;H)$ is dense too.

I want to show that if $V \subset H$ is a dense embedding then $L^2(0,T;V) \subset L^2(0,T;H)$ is dense too. Everything is a Hilbert space. Let $h \in L^2(0,T;H)$. Then $h(t) \in H$ for each $t$. By ...