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

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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 $T=0$?
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1answer
402 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 ...
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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, $v_i\perp v_j$ ...
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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
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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?
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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 ...
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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?
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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 ...
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2answers
95 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
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2answers
73 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 = ...
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1answer
444 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.
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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 ...
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1answer
54 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 $X^{\perp}$ ...
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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 ...
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0answers
34 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 ...
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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). ...
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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 ...
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1answer
315 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 ...
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1answer
67 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
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2answers
89 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 ...
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0answers
42 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 ...
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1answer
179 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.
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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 ...
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1answer
117 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 ...
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2answers
259 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 ...
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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 ...
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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 ...
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1answer
49 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 ...
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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 ...
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1answer
204 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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0answers
147 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 ...
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1answer
132 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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0answers
62 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 ...
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0answers
99 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 ...
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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 ...
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0answers
234 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 ...
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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 ...
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1answer
557 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 ...
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0answers
114 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 ...
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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 ...
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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 ...
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2answers
139 views

Extending bilinear form from subspace to whole space

Let $X$ be a linear subspace of a Hilbert space $Y$. Let $a(\cdot,\cdot):X \times X \to \mathbb{R}$ be bilinear. Suppose I know what $a$ is on $X$. Is there some theorem or other that tells me that ...
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0answers
45 views

Prove that the sequence is in $\ell^{2}$. [duplicate]

Let $(a_{n})$ be a sequence of complex numbers such that for every $(b_{n})\in \ell^{2}$the series $\sum_{1}^{\infty}a_{n}b_{n}$ converges. Prove that $(a_{n})\in \ell^{2}.$ What I've tried so far is ...
0
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1answer
132 views

Intersection of affine subspaces of finite codimension in Hilbert space

I'm wondering whether the following assertion is true: Any two affine subspaces of the same finite codimension in a ($\infty$-dimensional) Hilbert space either are parallel or have nonempty ...
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2answers
604 views

Matrix Representation of Operators in Infinite Dimensional (Separable) Hilbert Spaces

Suppose we have a separable Hilbert space (thus with a countable basis) and that we to represent an operator in matrix form, i.e: $$A: H \rightarrow H \\ \; \; \; \; \; \;x \;\rightarrow \sum_{j \in ...
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3answers
106 views

The Kernel of unbounded operator in Hilbert space

If $T$ is a densely defined operator from a subspace of a Hilbert space $H$ to a Hilbert space $K$, how to prove that $\mbox{Ker}(T)=\mbox{Ker}(T^*T)$?
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1answer
56 views

Does a cofinite dimensional subspace of a subspace remain cofinite dimensional upon taking closures?

Let H be a separable, infinite dimensional Hilbert space. Let X and Y be (not necessarily closed) subspaces such that X is a cofinite dimensional subspace of Y. Let X′ be the closure of X and Y′ the ...
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1answer
106 views

Tight Probability on Hilbert space

I am considering the following problem. Let $(X_j)$ be i.i.d. $N(0,1)$ random variables and $H$ a Hilbert space with orthonormal basis $(e_j)$. Let $$X:=\sum_j \frac{X_j e_j}{j}$$ And for any ...
2
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2answers
227 views

Unbounded operator $T $ is bounded below when $\overline T$ is bounded

How to prove the following? A densely defined symmetric operator $T$ in Hilbert space $H$ has a closure $\overline T$ which is bounded iff both $T,-T$ are bounded below (there exist constants $c,c' ...
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0answers
452 views

Closed unit ball in infinite dimensional normed linear space

I have to prove that in any infinite dimension normed linear space we have that the closed unit ball is not compact. I know that I have to construct a sequence such that $||x_n||=1$ and ...