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

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185 views

expansiveness imply “relaxed monotonicity”?

Let $(H, \langle \cdot, \cdot\rangle)$ be a real Hilbert space and let $T:H\rightarrow H$ be a map. If there exists a constant $h>0$ such that $$\|Tx-Ty\|\geq h\|x-y\|, \quad \forall x,y\in H,$$ ...
1
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1answer
78 views

Weak convergence in $L^2$ and CDF

Assume that for sequence $X_n \in L^2(\Omega,F,P)$ which converges in distribution to CDF $F_X$ ($F_n(t)\rightarrow F_X(t)$ for every point of continuity of $F_X$), we have also that $X_n$ converges ...
2
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1answer
82 views

Inner product on a von Neumann algebra

Let $M$ be a $\sigma$-finite von Neumann algebra (one which admits a faithful normal state) acting on a Hilbert space $H$. Denote its faithful normal state by $\omega$. We can define an inner product ...
5
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0answers
234 views

Hilbert spaces - equivalent norm

Let $H$ be a Hilbert space with a norm $\| \cdot \|_1$. Let $\| \cdot \|_2$ be another norm on $H$ which is equivalent with $\| \cdot \|_1$. It is easy to see that $(H, \| \cdot \|_2)$ is a Banach ...
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1answer
41 views

A question about a limit in a Hilbert space

Suppose $H$ is a Hilbert space, $v_{n},z \in H$ and suppose that $$ \lim\langle x, v_{n}\rangle = \langle x, z \rangle$$ for all $x$ in some dense subset of $H$. Then can I say that the sequence ...
2
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1answer
100 views

Equivalent inner product on Hilbert space

Let $(H, (\cdot,\cdot)_1)$ be a Hilbert space. Suppose also that $(\cdot,\cdot)_2$ is an inner product on $H$ which is norm-equivalent with $(\cdot,\cdot)$. Is it possible to write the second inner ...
3
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2answers
175 views

Representing the tensor product of two algebras as bounded operators on a Hilbert space.

Hi Math StackExchange, Let $A$ be a commutative, infinite dimensional, unital, *-algebra represented by bounded operators on a Hilbert space $H_A$. Next let $B$ be a finite non-commutative *-algebra ...
4
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2answers
238 views

a trace class operator problem

Could someone help me with this Prove that If $A$ and $B$ are positive trace class operators on a Hilbert space, then so is $A^zB^{(1-z)}$ for a complex number $z$ such that $0 <Re(z)< 1$. An ...
3
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2answers
233 views

Proof of the Riesz Representation Theorem

Theorem: Let $F$ be a continuous linear functional on the Hilbert space $H$, then $\exists !$ (exists one and only one) $y \in H$ such that $F(x) = (x,y)$ for $x\in H$. Proof: Uniqueness: ...
1
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1answer
69 views

Surjective function on product space

I know that, if $U$ and $V$ are closed subspaces of a Hilbert $(X,\langle\cdot,\cdot\rangle)$, then these statements are equivalents: $$i)\ U^{\perp}\subseteq U+V\quad\quad\quad ii)\ X = ...
2
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1answer
254 views

Find best approximation in Hilbert space $H = L_2[0; 1]$ for element $a(t)$

I try to solve this problem all day, but can't reach any progress in it. Can you give me some hints? Find best approximation in Hilbert space $H = L_2[0; 1]$ for element $a(t) = \frac{1}{\sqrt[4]t}$ ...
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1answer
285 views

Criteria of compactness of an operator

Suppose $K$ is a linear operator in a separable Hilbert space $H$ such that for any Hilbert basis $\{e_i\}$ of $H$ we have $\lim_{i,j \to \infty} (Ke_i,e_j) = 0$. Is it true that $K$ is compact? ...
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0answers
35 views

Partial completion of subspaces of Hilbert spaces

Assume $H$ is a Hilbert space and $H_1\subset H$ and $H_2 \subset H$ are (closed) subspaces with $H_1 \cap H_2 = \{0\}$. Is there an $H_3 \subset H$, such that $H = H_1 \oplus (H_2 \oplus H_3)$ ? ...
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1answer
146 views

Hilbert space proof

$X$ is a separable Hilbert space and $ A\in L(X,X)$ and compact. I need to prove that $A$ is approximately of finite dimension.
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0answers
38 views

a question in projection

Let $V=L^2(\Omega)$, and $$k=\{v \in V ~s.t ~||v||_{L^2(\Omega)}\leq 1 \}$$ I need to find projection for any $u \in V$ on $k$. Please help me.I do not have any idea about this problem. I have many ...
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1answer
1k views

Show that Y is a closed subspace of l2

This might be a straight forward problem but I wouldn't ask if I knew how to continue. Apologies in advance, I am not sure how to use the mathematical formatting. We are currently busy with inner ...
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1answer
1k views

Bounded sequence in Hilbert space contains weak convergent subsequence

In Hilbert space $H$, $\{x_n\}$ is a bounded sequence then it has a weak convergent subsequence. Is there any short proof? Thanks a lot.
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1answer
122 views

Why can I choose elements of $X$ (and $h \in L^2(0,T)$) in this way? (Dual spaces, norms, Bochner spaces)

This is from the book Vector Measures by Diestel and Uhl, page 98: Let $X$ be a Banach space. Let $\epsilon > 0$ and suppose first that $g = \sum_{i=1}^\infty x_i^* \chi_{E_i}$ where $x_i^* \in ...
5
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1answer
81 views

Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary?

Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary? What smoothness is required of the boundary? I would be grateful for some references to this.
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111 views

Concerning unbounded linear operators on a Hilbert space

Let $H$ be some Hilbert space and let $B:H\rightarrow H$ be a bounded linear operator and $T:H\rightarrow H$ an unbounded linear operator. Furthermore we assume that $T$ is closed ,i.e. it's graph in ...
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1answer
154 views

Closest point property of subset of Hilbert space - what are the conditions for infimum to be finite?

I am proving the closest point property of a subset of a Hilbert space $H$: given $h\in H$ and a closed, nonempty and convex subset $M\subset H$, consider $$d=\inf_{m\in M} \|m-h\|$$ I am trying to ...
3
votes
2answers
311 views

What is my operator norm (cannot get good enough bounds).

Given a space of square integrable functions $x(t)$ over the interval $[0;1]$ one can introduce a norm $$\|x(t)\|= \sqrt{\int_0^1 (x(t))^2 \, dt};$$ Then what is a norm of the transformation below ...
2
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1answer
204 views

Example for a sequence of operators converging pointwise, but not with respect to the operator norm

I am trying to understand the following example. Define $$T_n: l^2 \rightarrow l^2$$ $$T_n(x)=(0, ..., 0, x_{n+1}, ...).$$ It's rather clear that $T_n(x)$ converges for $0$ for every $x \in l^2$. ...
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2answers
232 views

Hilbert space with all subspaces closed

Does there exist an infinite-dimensional Hilbert space with all subspaces closed?
2
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1answer
303 views

Weak convergence in Hilbert space L2 implies convergence in distribution?

Does weak convergence in $L^2$ (for $X_n, X \in L^2$ we say that $X_n$ converges weakly to $X$ ($X_n \rightarrow^w X$) if for every $Y\in L^2$ we have $\mathbb{E}X_nY \rightarrow \mathbb{E}XY$) ...
0
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1answer
184 views

Can a Accumulation Point be an Eigenvalue?

I have a discrete (separable) infinite dimensional Hilbert Space with a compact operator defined on it. So 0 is an accumulation point (some theorem says so). Can 0 also be an eigenvalue? And how would ...
3
votes
1answer
120 views

How to prove $\mathcal{L}^2[(0,1)]$ is a Hilbert Space

Let $\mathcal{L}^2[(0,1)]$ denote the set of $\mathbb{C}$-valued square integrable functions on the interval [0,1]. Prove that $\mathcal{L}^2[(0,1)]$ forms a Hilbert Space. I believe that I can ...
2
votes
2answers
213 views

What's the connection between Banach/Hilbert spaces and tools like power series, Fourier transforms etc.

I've learned, abstractly, about Banach and Hilbert spaces, and more concretely about $l^p$ and $L^p$ spaces. I also understand these ideas have something to do with a variety of tools that are useful ...
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1answer
451 views

Prove a non-empty subset is closed in an inner product space

I hope someone would be able to help me with the finer details of this proof. Problem: $M$ is a non-empty set in an Inner Product Space (IPS) $X$. I need to show that the annihilator of $M$ which is ...
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1answer
67 views

Need explanation of problem in Temam (convergence, weak derivatives)

Let $V \subset H \subset V$ be Hilbert triple. We have $u_m$ is infinite differentiable from $[0,T]$ to $V$. Suppose $u_m \to u$ in $L^2(0,T;V)$ and $u_m' \to u'$ in $L^2(0,T;V^*)$ Suppose that it ...
2
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1answer
34 views

If $u_m \to u$ and $v_m \to v$, does $b(u_m,v_m) \to b(u,v)$?

In a Hilbert space $H$, if $u_m \to u$ and $v_m \to v$, does $b(u_m,v_m) \to b(u,v)$ if $b$ is a bounded bilinear form on $H$?
3
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1answer
854 views

Spectral Theorem for bounded compact, self-adjoint operators as corollary of Hilbert-Schmidt theorem

I'm following Debnath and Mikusinksi's "Introduction to Hilbert Spaces with Applications" and am trying to understand how the spectral theorem for compact self-adjoint operators is a corollary of the ...
2
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1answer
177 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
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1answer
380 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
164 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
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1answer
194 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 ...
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1answer
93 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 $A$
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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 ...
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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
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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 ...
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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 ...
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1answer
154 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
411 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?
3
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1answer
192 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
97 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 ...