All Questions
20 questions
1
vote
2
answers
60
views
Limit of sequence of non-negative, $C^0 \cap H^1$ functions that are zero at more and more points
I have a sequence $\{f_n\}$ of functions such that for each $n$, we got
$f_n \in H^1(\Omega) \cap C^0(\bar\Omega)$
$f_n \geq 0$
$\lVert f_n \rVert_{H^1(\Omega)} \leq C$ for a constant $C>0$ ...
- 475
0
votes
1
answer
53
views
Characterization of weakly compact operators
Let $X$ and $Y$ be normed spaces, I've read that if $T:X^*\to Y$ is a bounded linear operator such that $T^*(Y^*)\subset X$ then $T$ is weak*-to-weak continuous.
First question. Does $T^*(Y^*)\subset ...
- 149
1
vote
2
answers
176
views
Continuity and norm of operator on $l^2$
I need help with this:
Let $A$ be an operator on $l^2$, defined by
$$A(x)=y, \;x=(x_n)_{n \in N}, \; y = (\alpha_n x_n)_{n \in N}.$$ When is it continuous? Find its norm.
This is what I have done for ...
- 311
2
votes
1
answer
163
views
If $x$ is a càdlàg function and $f$ has compact support, how can we approximate $\sum_{s\in(a,\:b]}f(\Delta x(t))$?
Let $E_i$ be a normed $\mathbb R$-vector space and $x:[0,\infty)\to E_1$ be right-continuous. Assume $$x(t-):=\lim_{s\to t-}x(s)$$ exists for all $t\ge0$ and let $\Delta x(t):=x(t)-x(t-)$ for $t\ge0$.
...
- 13.9k
1
vote
1
answer
68
views
Check my proof: Does $F_k(u_n)$ belong to $W_0^{1, p}(\Omega)$?
Let $\Omega$ be an open bounded domain in $\mathbb{R}^N$ and let $(u_n)_n\subset W_0^{1, p}(\Omega)$ be a bounded sequence in $W_0^{1, p}(\Omega)$, $p>1$. Furthermore, fixed $k\in\mathbb{R}$, ...
user603537
3
votes
1
answer
241
views
Uniformly absolutely continuity: how to check it?
Let $(f_n)_n$ be a sequence. We say that it is a uniformly absolutely continuous sequence if given $\varepsilon>0$ there exists $\delta>0$ such that
$$\left|\int_{A} f_n\, \mathrm{d}\mu\right|&...
user603537
1
vote
0
answers
51
views
Map from $l^{\infty}$ to $l^{\infty}$ which includes Banach limit
Consider the following map:
$$ l^{\infty} \ni \{x_n\}_{n=1}^{\infty} \longmapsto \left\{x_n - \underset{j\rightarrow\infty}{\mathrm{Lim}}\ x_j\right\}_{n=1}^{\infty} \in l^{\infty},$$
where $\...
- 163
4
votes
3
answers
186
views
Why is it enough for $\lim\limits_{h \to 0} \vert f(x+h)-f(x)\vert = 0$ to show uniform continuity
I have seen an argument along the lines of:
if $f$ is a given function on $\mathbb R$ and, for any $x \in \mathbb R$, $\lim\limits_{ h \to 0}\vert f(x+h)-f(x)\vert=0$, then it immediately follows ...
- 1,838
1
vote
0
answers
55
views
Property takagi function by induction.
In a lecture on Applied Functional Analysis, the professor showed us some properties of the Takagi function from this paper. He wrote at the end the following property and said it could be easily done ...
- 403
2
votes
1
answer
827
views
If B is a subalgebra of A, conclude that B closure is a subalgebra of A
If B is a subalgebra of A, conclude that $\bar{B}$ is a subalgebra of A.
This is from Real Analysis by N. L Carothers chapter 12 exercise 3. The purpose of this is to lead up to the Stone Weierstrass ...
- 519
1
vote
1
answer
69
views
Defining equicontinuity in terms of sequences
Let $X$ be a metric space and let $\{f_n\}$ be a sequence of uniformly continuous functions on $X$. Is it true that if I can show that for any convergent sequence $x_n$, $x_n \rightarrow x$, there ...
- 1,664
6
votes
1
answer
253
views
convergence sequence and continuous functions
I have the following question: assume $C$ is a subset (we can assume it is convex and compact) of a Banach space $(X,\|\cdot\|)$, $f:C\longrightarrow C$ continuous and $x_{0}\in C$ a fixed point of $f$...
- 166
3
votes
1
answer
421
views
Norm operators bounded below implies almost uniform lower bound
I have a hard time proving (or disproving) the following statement about continuous linear operators:
$$(\exists c>0:\forall j:\|T_j\|\geq c)\Rightarrow(\exists\delta>0:\forall n:\exists x\in ...
- 1,203
0
votes
1
answer
42
views
Proving one of the properties of a strongly continuous semigroup
Let $X$ be a Hilbert space and $A\in\mathcal{L}(X)$ and $\displaystyle T(t)=e^{At}=\sum_{n=0}^{\infty}\frac{(At)^{n}}{n!}$.
I want to show that $T(t+\tau)=T(t)T(\tau)$.
So we have that $\...
- 1,068
0
votes
1
answer
52
views
Please verify my work about an equicontinuous sequence
Please check this work below. It is self-explanatory. I am unsure because I use a sequence composed with another sequence with the same index ($f_n^{-1}(u_n)$).
We have a sequence of functions $f_n:\...
- 771
0
votes
1
answer
295
views
Are these $f_n$ equicontinuous?
Let $f_n$ be a sequence of real-valued functions defined on $\mathbb{R}$ satisfying
$f_n \to f$ uniformly in the compact subsets of $\mathbb{R}$
$f_n^{-1}$ is bi-Lipschitz
$1 \leq (f_n^{-1})'(x) \leq ...
- 771
3
votes
1
answer
730
views
$x_n$ convergence to $x$ implies $f_n(x_n)$ convergence to $f(x)$. prove that $f$ is continuous
Let $f$ and $f_n$ be functions from $\mathbb{R} \rightarrow \mathbb{R}$ Assume that $f_n (x_n) \rightarrow f (x)$ as $n\rightarrow \infty$ whenever $x_n \rightarrow x$. Prove that $f$ is
continuous. ...
- 2,248
0
votes
1
answer
251
views
Proof of a special case of Banach's fixed point theorem
I have to prove the following special case of the theorem:
Let $f : I \to I$ be Lipschitz continuous on the closed (not bounded) interval $I=[0,\infty)$ with Lipschitz constant $L \lt 1$. Then $f$ ...
- 1,367
2
votes
1
answer
78
views
Continuity of $x+y$ and $xy$ in $\mathbb{R}^{\infty}$
How can I show (or where can I find) that in $\mathbb{R}^\infty$: $f(\textbf{x},\textbf{y})=\textbf{x}+\textbf{y}$, $g(\textbf{x}, k)=k\cdot \textbf{x}$ are continuous functions?
($g$ is from $\mathbb{...
- 303
0
votes
1
answer
475
views
Maps sending weakly convergent sequences to weakly convergent sequences are continuous?
Well, the question is in the title. I understand that they are continuous in the weak topology, but can't see that it must hold for the norm topology.
Please help me.
- 1,073