3
votes
1answer
43 views

Are translates of Gaussians an overcomplete set in $L^2(\Bbb R)$?

Consider the Gaussian $\exp(-t^2/2)$. Is it the case that any function in $L^2(\Bbb R)$ can be written as a limit of a sum of scalings and translations of Gaussians? That is, for any $f\in L^2(\Bbb ...
2
votes
2answers
32 views

Closed subspaces of $L^2(0,1)$

I would like to prove that the almost-everywhere constant functions, and the functions whose integral is 0 are closed subspaces of $L^2(0,1)$. It's readily seen that they are subspaces. I'm finding ...
0
votes
0answers
28 views

Tensor Product of Hilbert Spaces: incomplete?

Let $\mathcal{H}$ be an infinite dimensional Hilbert space and $\mathcal{H}\otimes_0\mathcal{H}$ its algebraic twofold tensor product. Define a scalar product on it as ...
1
vote
1answer
33 views

Convergence of a sequence of unit vectors in a Hilbert space

Let $E$ be a vector space (over $\mathbb{R}$) with a positive definite hermitian form and let $\{x_{n}\}$ ($x_{n} \not= 0$) be a sequence converging to $x$ in the $L^{2}$ norm ...
5
votes
1answer
81 views

Strong convergence of an “averaging” operator

Let $X$ be an Hilbert space and $S:X \rightarrow X$ be a bounded linear operator with $||S||=1 $ Define $$T_n= \frac{1}{n} \sum_{r=0}^{n-1} S^r$$ I want to show it converges strongly to some ...
1
vote
1answer
39 views

Completeness: $\mathcal{l}^2(S)$

Surely, for countable index sets this is just the diagonal trick: $\#S<\infty$ However for arbitrary index sets how do I prove that the limit will actually have only countable non vanishing ...
1
vote
0answers
18 views

Limit of function of an operator

Let $A_n$ be a sequence of bounded, self-adjoint operators on Hilbert space $\mathcal{H}$. Let us assume that for some vector $\psi\in\mathcal{H}$, $$\lim_{n\rightarrow\infty}A_n\psi = \alpha ...
2
votes
2answers
66 views

Convergence of sums using Hilbert space techniques [duplicate]

Let $a_n$ be a sequence of real numbers such that $\sum_{n=1}^{\infty} a_nb_n < \infty$ for any sequence $b_n$ satisfying $\sum_{n=1}^{\infty}b_n^2 < \infty$. Prove that ...
1
vote
1answer
35 views

Weak convergence of subsequence in Hilbert spaces

Prove that if $x_n$ is a sequence in $H$ (Hilbert space) with $\sup_n||x_n||\le1$, then there is a subsequence $\{x_{n_j}\}$ and an element $x$ of $H$ with $||x||\le 1$ such that $x_{n_j}$ converges ...
2
votes
1answer
77 views

Two questions on Banach-valued spaces of integrable functions

Let $V$ be a Banach space with dual $V'$ and suppose that $V$ is included into the Hilbert space $H$ so that the inclusion is continuous and dense. Then after identification $H = H'%$ we have $V ...
5
votes
0answers
69 views

Does this sequence of operators converge in norm or strongly?

Let $H$ be a Hilbert space and $\mathcal{L}(H)$ the set of all bounded linear operators $L:H\to H$, equiped with the usual norm $\|\cdot\|_{\mathcal{L}}$. Let $T:D(T)\subset H\to H$ be a ...
1
vote
0answers
47 views

Product of strong and weakly converging sequences

Consider a sequence of functions $\{u_n\}\in L^2([0,T],L^2(\Omega)) $ which converges strongly to a function $u\in L^2([0,T],L^2(\Omega))$. Then $u_n \rightarrow u \;\; a.e. $ in ...
1
vote
0answers
52 views

Difference between unconditional and absolute convergence in Banach spaces

One can show that in any finite-dimensional normed vector space absolute convergence is equivalent to unconditional convergence. It's not hard to show that if we have an orthonormal sequence in ...
0
votes
0answers
59 views

Unconditionally convergence in Hilbert space

Let $H$ be a complex Hilbert space and $e_1, e_2,...$ be a countable orthonormal system in $H$, $c_1, c_2,...$ is a sequence of complex numbers. How to prove that if $c_n$ is square-summable then ...
1
vote
1answer
67 views

Weak convergence equal to coordinate-wise convergence

Show for the Hilbert Space $\ell^2(\mathbb{N})$ weak convergence of a bounded sequence $(x_k)_{k \in \mathbb{N}}$ is equal to coordinate-wise convergence. A sequence $(x_k)$ is weak ...
0
votes
1answer
74 views

absolutely convergent series in Hilbert space

Is it possible to find an infinite dimensional Hilbert space, where every convergent series is absolutely convergent? I could not find any clue to find an example of such type or to disprove that. ...
0
votes
1answer
215 views

A problem about the strong convergence in a Hilbert space.

Let $X_n$ be a sequence in a Hilbert space H. If $\| X_n \| \to \| x \|$, show that $X_n$ converges strongly to $x$. Note that I'm not saying anything about weak convergence.
0
votes
2answers
112 views

Subspace of $C^1 [0,1]$

Consider the inner product space of continuously differentiable functions, $C^1 [0,1]$ with inner product:$$\left<f,g\right> =\int_{0}^1f(x)\overline{g(x)}\,dx + ...
1
vote
1answer
516 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 ...
2
votes
1answer
81 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
77 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
189 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 ...
3
votes
1answer
80 views

How to show projection of $L^2$ function converges to that $L^2$ function

My teacher said that if $P_n f = \sum_{j=0}^n(f,w_j)w_j$, where $w_j$ is orthonormal basis of $L^2$, then $|P_n f- f|_{L^2} \to 0$ for $f \in L^2$. How do I prove this? I thought $$|P_nf - f| = ...
0
votes
2answers
227 views

Weakly convergent sequence

Consider a sequence $(x_n)_n$ in Hilbert space $H$ such that $\langle x_m,x_n\rangle=\delta_{mn}$ where $\delta_{mn}$ equals one if $m = n$ and $C$ otherwise. Prove that $(x_n)_n$ is a weakly ...
2
votes
1answer
46 views

Inequality involving a sequence in Hilbert space

Let $H = L^2(0,T;V)$ where $V$ is separable in $H$. We have $y_n(t) = \sum_{i=1}^n c_{i,n}(t)b_i$ where $b_i$ are basis vectors in $V$. Suppose we have the estimate $$\lVert y_n \rVert_H^2 = \int_0^T ...
1
vote
1answer
327 views

In a separable Hilbert space, how to show that the orthogonal projection onto a subspace of $n$ orthonormal basis elements converge?

Could anyone help me with this problem? I don't know where to start. Let $\{ e_n \}_{n=1}^\infty$ be an orthonormal basis in a separable Hilbert space $H$. Denote by $P_n$ the orthogonal ...
6
votes
2answers
2k views

Weak Convergence implies boundedness and componentwise convergence

Let $\ell^2$ be the set of real number sequences $\{a_n\}$ such that $\sum a_n^2 <\infty$. Let $\langle a_n,y\rangle \rightarrow \langle a,y\rangle$ for some $a\in \ell^2$ and for all $y\in ...
1
vote
0answers
234 views

Diagonal Dominance and Spectral Radius

For positive semi-definite matrices, $A$ and $B$ with real entries, Let: $X=I-(2Diag(A)-B)^{-1}(A-B)$ The spectral radius $\rho(X) \leq ||X||$. As, $(2 Diag(A)-B)$ becomes a better approximation ...
1
vote
2answers
139 views

Cauchy+pointwise convergence $\Rightarrow$ uniform converges (for an operator in a Hilbert space)

Suppose that the sequence of operators in a Hilbert space $H$, $\left(T_{n}\right)_{n}$, is Cauchy (with respect to the operator norm) and that there is an operator $L$, such that ...
2
votes
2answers
462 views

Weak convergence in Hilbert spaces

Definition of the problem Let $\mathcal{H}$ be a Hilbert space, and let $\left(x_{n}\right)_{n\in\mathbb{N}}\subset\mathcal{H}$ be a sequence. Prove the following: If ...
0
votes
2answers
106 views

Net of Projections

I am trouble proving the following proposition in Conway's functional analysis book. $H$ is an arbitrary Hilbert space, $I$ is an index set. Prop - Let $\{P_i:i\in I\}$ be a family of pairwise ...
4
votes
1answer
391 views

Weak convergence

Let $H$ be a Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and let $V,W$ be two closed subspaces. For $x_0\in H$ we may define the sequence of projections $$x_{2n+1}=P_W(x_{2n}), \qquad ...
2
votes
1answer
144 views

A question on norm of error vector

Let $(s_n)_{n \in \mathbb{N}}\in\ell^2(\mathbb{N})$ (i.e. $\displaystyle \sum_{n=0}^{\infty}\vert s_n\vert^2<\infty$). Define vectors $A=[A_1,\ldots,A_M]$ and $B=[B_1,\ldots,B_M]$ with coordinates ...
4
votes
3answers
357 views

Convergence of a sequence of periodic functions

Motivated by the homogenization theory which studies the effects of high-frequency oscillations in the coefficients upon solutions of PDE, I am thinking about the following question. Let the ...