Tagged Questions
2
votes
1answer
53 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
$$
u^{k+1}=(1-\alpha)u^k+\alpha ...
1
vote
1answer
73 views
Need help in showing that $F(x)/x^{1/q}$ goes to $0$ as $x$ goes to $0$ and $\infty$.
$1<p<\infty$, $f\in L^{p}(0,\infty)$, $p^{-1}+q^{-1}=1$, define $$F(x)=\int_{0}^{x}f(t)dt,$$ then I need to show that $\frac{F(x)}{x^{\frac{1}{q}}}\rightarrow 0$ as $x\rightarrow 0$ and ...
2
votes
2answers
28 views
Limit of bounded operators
Suppose $T_n$ is a sequence of self-adjoint bounded operators on a Hilbert space, and $T_n \rightarrow T$ in operator norm, $T$ being also bounded and self-adjoint.
Do we then have: $T_n^m\rightarrow ...
3
votes
2answers
63 views
Do $L^2$ convergence and continuity imply pointwise convergence?
It is said here that $L^2$ convergence and continuity imply pointwise convergence (just before paragraph $5.2$) but I can't find how to prove it. Does anyone see how ?
2
votes
0answers
63 views
When $\ell^2$-convergence implies $\ell^1$- convergence?
Consider a sequence $(x_n)_{n\in\mathbb N}$ in $\ell^1$ (sequences taking their values in $\mathbb R$), where $x_n=(x_{i,n})_{i\in\mathbb N}$.
What are sufficient conditions on the sequence ...
1
vote
1answer
22 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 ...
0
votes
1answer
37 views
Convergence in normed spaces
I consider a sequence of elements $\{f_n\}_n$ in a normed space $X$ such as $\Vert f_n-f\Vert_X\to 0, n\to\infty$. Let $\{Q_n\}_n$ a complex sequence with $Q_n\to Q$ as $n\to\infty$.
Does $Q_nf_n$ ...
1
vote
1answer
39 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 ...
1
vote
0answers
45 views
weakly convergent sequence in $l^1$ [duplicate]
Prove that every weakly convergent sequence in $l^1$ converges.
By Riesz on $L^p$ spaces, every linear functional $L\in (l^1 )^*$, is $L(x) = \langle u,x\rangle$ for some $u\in l^{\infty}$ ...
1
vote
1answer
39 views
Convergence in norm operator topology
I have to prove that a sequence $A(\varepsilon)$ of operators between Hilbert spaces $A(\varepsilon):H_1\to H_2$ converges, when $\varepsilon\to 0^+$, to an operator $B:H_1\to H_2$ in the uniform norm ...
3
votes
2answers
62 views
Extracting a subsequence from a sequence of $\mathcal{L}^1$ functions
Any help with the following problem is appreciated.
Given: a sequence of nonnegative functions $(g_n)$ which are U.I. (uniformly integrable) in $\mathcal{L}^1(0,1)$ with $\sup_n \Vert g_n \Vert_1 ...
2
votes
1answer
18 views
WOT convergence of the operators
Let $\{A_n\}$ and $\{B_n\}$ be sequences in $\mathfrak{B}(H)$ such that $A_n \to A$ in WOT and $B_k \to B$ in SOT. Show that $A_nB_n \to AB$ in WOT.
This is from Conway's book "A Course in Funcional ...
0
votes
1answer
58 views
Open or closed set of converging sequence
I have come across a question regarding whether a set is open or closed... Needing to give details of why its open or closed..
$C_0 $ the set of sequences converging to $0$ in $(\ell^{\infty} ...
2
votes
1answer
31 views
Convergence theorems for $\mathcal{L}^p$ spaces
Do the MCT, DCT and Fatou's lemma extend to $\mathcal{L}^p$ spaces also for $ 1 \leq p \leq \infty$. If they do not what are cases that they would fail and why?
Any help with this question is ...
0
votes
1answer
25 views
Weak and normwise convergence of sequence of linear functionals
Is this sequence of linear functionals weakly (normwise) convergent : $$f_n((x_j))=\sum_{k=1}^{n}{\frac{x_k}{k}} , (x_j) \in \ell_1\,?$$
0
votes
1answer
103 views
How to show this sequence of functions weakly converge?
$X = C[0,1]$. $$x_n(t) =
\begin{cases}
nt, & \text{for $0 \leq t \leq \frac{1}{n}$ } \\
2-nt, & \text{for $\frac{1}{n} \leq t \leq \frac{2}{n}$ } \\
0, & \text{for $\frac{2}{n} \leq t ...
0
votes
0answers
59 views
Exponential form of the selfadjoint operator
Let $A$ be a self adjoint operator on a Hilbert space $H$. Let $t \geq 0$.
Show that $\left<x, \left(I+\frac{1}{n}A \right)^{tn}y \right> \to \left< x, e^{tA}y \right>$ as $n \to \infty$.
...
2
votes
2answers
53 views
Total sets in Banach spaces.
We call the set total if its linear span is dense in a given normed space.
Let $X$ be a Banach space and let $D$ be a total set in $X$.
For the sequence $\{T_n\}$ of bounded operators on $X$. we ...
0
votes
0answers
39 views
Limit with theorem of dominated convergence
Let $f\in L^{2,s}(\mathbb{R}^3)=\bigg\lbrace u\bigg|\int_{\mathbb{R}^3}dx\,u(x)(1+|x|^2)^s<\infty\bigg\rbrace$ ($s>\frac{1}{2}$)
I have to calculate this limit
$$\lim_{|x-y|\to ...
2
votes
2answers
115 views
$C^\infty(R^n)$ is a Banach Space when equipped with topology of uniform convergence
Prove $C^\infty(\Bbb R^n)$ is a Banach Space when equipped with topology of uniform convergence.
$C^\infty(\Bbb R^n)$ is space of all continuous functions that converge to $0$ at $\infty$.
And, the ...
1
vote
2answers
60 views
Operator norm converging to 0 for certain condition
Let $X$ be a finite-dimensional normed space and $T_n : X \to X$ a sequence of linear operators such that $\lim_nT_nx = 0$ for all $x$ in $X$. Prove that $\lim_n\|T_n\|=0$.
2
votes
0answers
40 views
Limit in norm of a Sobolev space
I consider, for $s>\frac{1}{2}$, the space $L^{2,-s}(\mathbb{R}^3)=\bigg\{f: \int_{\mathbb{R}^3}|f(x)|^2(1+|x|^2)^{-s}<\infty\bigg\}$ and I have to show that the function ...
1
vote
1answer
42 views
Convergence in norm operator
If I have an operator valued functions $A(z):H_1\to H_2$ such that the following limit
$$\lim_{z\to z'}A(z)=A(z')$$
exists in the uniform topology of $B(H_1,H_2)$, that is
$$\Vert ...
1
vote
1answer
59 views
Regarding an isomorphism between a subspace of $\ell^{\infty}$ and $\ell^1$
Let $c_0$ be the subspace of $\ell^\infty$ consisting of sequences that converge to $0$. Show that $c_0$ is a closed subspace of $\ell^{\infty}$ whose dual space is isomorphic to $\ell^1$. Conclude ...
1
vote
0answers
51 views
Dense subset of the conjugate space
The question is:
Let $X$ be a normed linear space and let $B$ be a dense subset of $X^*$(the conjugate space of $X$). If a sequence $\{x_n\}$ in $X$ is bounded, and if $\lim_n x^*(x_n)$ exists for ...
2
votes
1answer
36 views
Strong convergence of multiplication operator
I am looking for a necessary and sufficient condition for a sequence of multiplication operators $T^{(k)}$ to converge to zero strongly. (i.e. $\forall x \in \mathcal{H} \quad ||T^{(k)}x - 0|| \to 0$ ...
1
vote
2answers
73 views
From weak and weak star to norm convergence
I haven't found this yet and I'm somehow not sure if my idea is correct.
The Problem: Let $X$ be a separable Banach-Space, let $x_k\to x$ weakly and such that for every $\lambda_k \to \lambda$ ...
1
vote
1answer
50 views
Sequential continuity in normed linear spaces
I am trying to prove the following "contiuity-type" result.
Let $X,Y$ normed linear spaces. Let $\{T_n\} \to T \in \mathcal{L}(X,Y)$ and $\{u_n\} \to u \in X$. Show that $\{T_n(u_n)\} \to \{T(u)\} ...
1
vote
1answer
61 views
Sequences in Banach spaces
I am very bad with proofs that ask you to show that "$\exists$ something..." because in most of them you have to explicitly show the something. The following question is one such. Any help will be ...
1
vote
2answers
114 views
A sequence converging weakly in $\ell^p$, for $p >1$ and failing to converge weakly for $p=1$
For $1 \le p < \infty$ and each index $n$, let $e_n \in \ell^p$ have $n$-th component 1 and all other componenets $0$. I want to show that $p>1 \Rightarrow \{e_n\} \to 0$ weakly in $\ell^p$ and ...
0
votes
1answer
51 views
Failure of convergence to 0
Consider the interval $I = [0,1]$ and the sequence of functions:
$$f_n(x) = (-1)^k \ \text{for} \displaystyle \frac{k}{2^n} \le x < \frac{k+1}{2^n} \ \text{where} \ 0 \le k < 2^n - 1$$
I want ...
1
vote
0answers
53 views
A question about convergence in $L^p$. [duplicate]
Let $E$ be measurable and $1 \le p \le \infty$. Suppose $\{f_n\}_{n \in \mathbb{N}}$ all measurable and $\{f_n\}_{n \in \mathbb{N}} \to f$ pointwise a.e. $E$. For $p$ as above, I want to show that:
...
2
votes
2answers
83 views
Weak limit of disjoint normalized sequence in $L^p$
I want to prove that the weak limit of a disjoint normalized (pairwise disjoint supports, elements of norm $1$) sequence $(f_n)$ in $L^p$ for $p >1$ is zero ?
I started with the measure of ...
3
votes
1answer
71 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
$$|P_nf - f| = ...
0
votes
1answer
175 views
About Banach Spaces And Absolute Convergence Of Seires
How to prove the following two assertions:
If in a normed space $X$, absolute convergence of any series always implies convergence of that series, then $X$ is a Banach space.
In a Banach space, ...
1
vote
0answers
155 views
Rademacher function and weak convergence
The function $r_{n}:[0,1]\rightarrow \{-1,1\}$ be defined by $r_{n}(t)=\operatorname{sgn}(\sin(2^{n}\pi t))$ is known as the $n$-th Rademacher function.a) Show that $r_{n}\xrightarrow{w}0$ in ...
1
vote
1answer
140 views
Show $\mathbb R^n$ is complete.
Show $\mathbb R^n$ is complete.
At this point, I am trying to work through the problem in my textbook, there is one step that I do not understand and would like explained. Here's my proof so far:
...
5
votes
2answers
172 views
Unit ball in $C[0,1]$ not sequentially compact
This question is taken from Saxe K -Beginning Functional Analysis.
Show that the closed unit ball in $C[0,1]$ is not compact by proving that it is not sequentially compact.
(It's assumed that we ...
2
votes
1answer
63 views
Convergence in norm independent of the choice of the norm.
When I read the proof of the Lie product formula in Reed Simon's book on functional analysis (which essentially reduces to showing $\left\Vert X_n - Y_n\right\Vert\rightarrow 0$ as $n\rightarrow 0$ ...
2
votes
2answers
77 views
A sequence in $C([-1,1])$ and $C^1([-1,1])$ with star-weak convergence w.r.t. to one space, but not the other
The functionals
$$
\phi_n(x) = \int_{\frac{1}{n} \le |t| \le 1} \frac{x(t)}{t} \mathrm{d} t
$$
define a sequence of functionls in $C([-1,1])$ and $C^1([-1,1])$.
a) Show that $(\phi_n)$ converges ...
2
votes
2answers
70 views
Inequality regarding norms and weak-star convergence
Let $X$ be a normed space and $(x'_n) \subseteq X'$ a sequence of functionals where $x_n'$ has $x'$ has its limit in the *-weak topology in $X'$. Show that
$$
||x'|| \le \operatorname{lim inf}_{n\to ...
8
votes
2answers
184 views
Compact maps problem in Lax
In Functional Analysis of Peter Lax there are the following exercise
Show that if $\bf C$ is compact and $\{{\bf M}_n \}$ tends strongly to $\bf M$, then $\bf CM_n$ tends uniformly to $\bf CM$.
...
6
votes
1answer
88 views
How many well-behaved norms are there on $C[0,1]$?
Let $\Vert \cdot \Vert$ a norm on $X=C([0,1])$ s.t.
1. $X$ is complete w.r.t. $\Vert \cdot \Vert$;
2. convergence in $\Vert \cdot \Vert$ implies pointwise convegence, i.e.
$$
\Vert ...
4
votes
2answers
121 views
Do weak convergence and convergence of norms imply convergence in $L^2$?
Let $(f_n)_n\subseteq L^2(0,1)$ s.t.
$$
f_n \rightharpoonup f, \qquad\qquad \Vert f_n\Vert_2 \to \Vert f\Vert_2
$$
where $\rightharpoonup$ means weak convergence. Is it true that $f_n \to f$ ...
1
vote
1answer
95 views
Normal convergence
I have some problems to apply normal convergence of series of functions in any vector space.
In fact $(f_{n})$ is a sequence of differentiable functions defined from a topological space $X$ to a ...
1
vote
1answer
93 views
The convergence of the adjoint operator
If a sequence of operator $A_n$ converges in norm to $A$, i.e. $\lim \lVert A_n-A\rVert=0$)where $A_n$ and $A\in B(H)$ ($H$ is the Hilbert space). Is it true that $A_n^*$ converges in norm to $A^*$?
0
votes
2answers
145 views
Weakly convergent sequence
Consider a sequence $(x_n)_n$ in Hilbert space $H$ such that $\langle x_m,x_n\rangle=\delta_{mn}$ where $\delta_{mn}$ equals one if $m = n$ and $C$ otherwise. Prove that $(x_n)_n$ is a weakly ...
3
votes
1answer
75 views
$f_k \rightarrow f$ in $L^p$ and that $g_k \rightarrow g$ weakly in $L^q$. Show that $f_k g_k \rightarrow fg$ weakly in $L^1$
I want to solve the following exercise, and I thankfully welcome some hints. Note that this is not homework.
Problem: Let $1 < p,q < \infty$ be conjugate exponents. Assume $f_k \rightarrow ...
1
vote
1answer
135 views
Convergence in $\ell^p$ norm provided it weakly converges.
I need some help with the following problem :
$1<p < \infty$ , let $x_n$ be a sequence in $\ell^p$ and also $x\in \ell^p$ . I am interested in showing $$\lim_{n\to \infty} \|x_n-x\|_p\to0$$ ...
3
votes
3answers
106 views
Is the space $C[0,1]$ locally compact?
Let $C[0,1]$ be the space of continuous functions, equipped with the $\sup$-norm.
My question is, how one can prove, that this space is not locally compact? Is it possible to show this explicitly, by ...
