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

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3
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1answer
535 views

A strictly positive operator is invertible

Suppose that $H$ is an Hilbert space, and $T: H \to H$ is a self-adjoint strictly positive operator (i.e. $\langle Tx,x\rangle > 0$ for all $x \neq 0$). How do I show that this operator is ...
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1answer
231 views

Operator self-adjoint

I have this paragraph : "Let M be a Hilbert-Riemannian manifold. $f \in C^2(M,R), p \in K$ is called a nondegenerate critical point, if $d^2 f (p)$ has a bounded inverse. Since $A = d^2 f (p)$ is a ...
3
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0answers
77 views

Is this operator bounded?

Let $w_j$ be a basis ( not orthogonal) of the Hilbert space $H$. For $h = \sum^\infty a_iw_i$ define $P_n(h) = \sum_{i=1}^n a_iw_i$. Is this operator bounded in $H$ I don't think it is but I feel ...
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2answers
367 views

Projection operator for non-orthonormal basis

Let $V \subset H$ be Hilbert spaces. Let $\{v_j\}_{j=1}^\infty$ be a basis for $V$ and $H$. Define $V_N$ to be the span of $\{v_j\}_{j=1}^N$. We can define a projection operator $P:H \to V_N$ by ...
2
votes
1answer
87 views

Does a continuous embedding preserve gaps between subspaces?

I have a separable, reflexive Banach space $(V,\|\cdot\|)$ that is continuously and densely embedded in a Hilbert space $(H,|\cdot|)$. This means, there is a bounded linear injection map $j\colon V ...
3
votes
1answer
85 views

How to prove that the set $e^{inx}$ is closed with respect to this measure?

Why is the set $\{e^{inx}\}$ closed in $L^2(d\nu)$, where $d\nu(x)=(1+|h(x)|)dx+|ds(x)|$? $d\nu$ is defined in this proof I am struggling to understand, where $\mu$ is a complex Borel measure on ...
3
votes
2answers
91 views

What is the correct definition of the cuspidal subspace of $L^2$?

I have a few (semi-)related questions regarding certain Hilbert space representations of locally compact groups that come up in the theory of automorphic forms. Let $G$ be a unimodular locally ...
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1answer
72 views

An operator on $H\times H$, with $H$ Hilbert

Let $(H, \langle \cdot,\cdot\rangle_H)$ a Hilbert complex space and consider $H\times H$ with the inner product $$\langle (u,v),(z,w)\rangle_{H\times H}\ =\ \langle u,z\rangle_H + \langle ...
2
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0answers
315 views

Direct sum of Hilbert spaces

Let $H$ be a separable Hilbert space with an orthonormal basis $\{e_n\}_{n =0}^{\infty}$. Consider a direct sum, $H \oplus H$. What is the orthonormal basis of $H \oplus H$ ? Is it $(e_n, e_m)_{n,m ...
3
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1answer
252 views

Proving Inner Product Space

Let $E=C^1 [a,b]$ be the space of all continuously differentiable functions. For $f,g \in E$ define $$ \langle f,g \rangle \ = \ \int_a^b f'(x) \ g'(x) \ dx$$ Is $\langle f,g \rangle$ an inner product ...
2
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0answers
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 $$\|Tx-Ty\|\geq h\|x-y\|, \quad \forall x,y\in H,$$ ...
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vote
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
votes
1answer
81 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
222 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 ...
0
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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
votes
1answer
94 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
votes
2answers
166 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
votes
2answers
226 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
votes
2answers
227 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: ...
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vote
1answer
68 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
votes
1answer
242 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 $a(t) = \frac{1}{\sqrt[4]t}$ ...
10
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1answer
276 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
145 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 ...
3
votes
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
votes
1answer
80 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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0answers
108 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
146 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
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2answers
304 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
193 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 $x \in l^2$. ...
5
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2answers
226 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
296 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
119 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
199 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
448 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 ...
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votes
1answer
31 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
votes
1answer
775 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
votes
1answer
161 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
372 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
161 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 ...
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1answer
184 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
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
92 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
106 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 ...