All Questions
41 questions
2
votes
1
answer
68
views
Limit of sequence in $C_b(\mathbb{R})$ [closed]
Consider $C_b(\mathbb{R}) = \{f| f:\mathbb{R}\rightarrow \mathbb{R}, f \text{ is continuous and bounded}\}$ with norm $\lVert\cdot\rVert_\infty$. For $f\in C_b(\mathbb{R})$ define $\displaystyle g_n(x)...
- 161
2
votes
0
answers
48
views
Is this infinite sum of matrices convergent?
For a matrix $A$, by $||A||$, I mean the matrix-norm induced by the $\ell^2$-norm. Let $A\in\mathbb{R}^{m\times m}$, $B\in\mathbb{R}^{m\times p}$, $C\in\mathbb{R}^{n\times m}$ with $||A|| < 1$. ...
-1
votes
1
answer
65
views
Prove Weak convergence in $H$.
Let the sequence $(x_n)$ of points of the Hilbert space $H$ weakly coincide
to the point $x ∈ H$. Prove that for an arbitrary point $y ∈ H-\{x\}$ has
the place is uneven
$\lim_{n→∞}||x_n − x|| < \...
0
votes
2
answers
52
views
Prove that $\exists l\in\mathbb R, f(x,y)\rightarrow l$ when $(x,y)$ goes to $(0,0)$
Let $f:\mathbb R^*\times\mathbb R\rightarrow \mathbb R $,
$\forall \phi:\mathbb R^* \rightarrow \mathbb R$ such that $\phi(x)\rightarrow 0$ when $x$ goes to $0$. When $x$ goes to $0$ :
$$\exists l\in\...
1
vote
1
answer
109
views
Distance from $(1,0,0,...)$ to closed subspace in $l^2$
can someone help me with this problem?
Find distance $d(x,L_n)$ in $l^2$ from $x=(1,0,0,...)$ to closed subspace $L_n=\{x \in l^2: x=(x_1,x_2,...), \sum_{k=1}^n x_k = 0\}$. Also, find $\lim_{n \to \...
- 311
0
votes
1
answer
78
views
Definition of sequentially compact subset
I am reading "Beginning Functional Analysis" by Karen Saxe, and I have came upon the definition of a sequentially compact subset:
Let $(M,d)$ be a metric space. A subset $E \subset M$ is ...
- 147
3
votes
1
answer
74
views
stronger variant of "$l^p$ spaces are increasing in $p$"： $\bigcup_{1\leq p\lneq q}l^p\neq l^q$
Problem
Let $\mathbb{K}$ be either $\mathbb{R}$ or $\mathbb{C}$. Let $x\in\mathbb{K}^\mathbb{N}$ be any sequence in $\mathbb{K}$. Let the $p$-norm be defined by $$\lVert x\rVert_p:=\left(\sum_{n\in\...
- 631
1
vote
3
answers
334
views
$l^p$ spaces are increasing in $p$
Problem
Let $\mathbb{K}$ be either $\mathbb{R}$ or $\mathbb{C}$. Let $x\in\mathbb{K}^\mathbb{N}$ be any sequence in $\mathbb{K}$. Let the $p$-norm be defined by $$\lVert x\rVert_p:=\left(\sum_{n\in\...
- 631
0
votes
0
answers
31
views
Correctness of method - if $\{ y_n \} $ is unbounded, prove $\{ y_n z_n\} $ does not converge to $0$
Problem: Let $c_0$ be the space of complex sequences that converge to $0$. Let $\{ y_n\}_{n \in \mathbb{N}} $ be a sequence of scalars such that for arbitrary sequence $\{ z_n\} \in c_0 $ we have: $...
- 517
2
votes
3
answers
116
views
Show that sequence must be in $ℓ^\infty$
Given a sequence $(a_n) \in \mathbb{R}$ so that for every sequence $(x_n) \in c_0 : (a_nx_n) \in c_0$. Show that this implies that $(a_n)$ has to be in $ℓ^\infty$.
My thoughts: $(a_nx_n) \in c_0$ ...
0
votes
1
answer
67
views
Sequences and little-o notation
Let $(X, \|\cdot\|)$ be a Banach space and $J\in\mathcal{C}^1(X, \mathbb{R})$ be a functional. Let $(u_n)_n\subset X$ be a Palais-Smale sequence at level $c\in\mathbb{R}$, i.e.
$$J(u_n)\to c\quad\mbox{...
user603537
-1
votes
1
answer
42
views
Convergent subsequence in $C^1$
We consider two closed interval in $\mathbb{R}$ : $[a,b]$ and $[c,d]$. Moreover, let us consider a sequence $\{f_{n}\}_{n = 1}^{\infty}$ uniformly bounded in $C^1([a,b]\times [c,d])$. My question is: ...
- 21
0
votes
1
answer
79
views
Find $\varlimsup_{x \rightarrow + \infty} (\cos\sqrt{x-2015}-\cos\sqrt{x+2015})$
Find $$\varlimsup_{x \rightarrow + \infty} (\cos\sqrt{x-2015}-\cos\sqrt{x+2015})$$
I don't understand what the overline means. I think about upper limit of the sequence, but I'm not sure that's right ...
- 479
0
votes
1
answer
53
views
Norm determines lower bound for invertible elements
Let $A$ be some unital Banach *-algebra. M. Takesaki, Theory of Operator Algebras, vol. 1. Springer, 1979. claims that if $|\lambda|>r(x):=\lim\inf_{n\rightarrow\infty}\|x^n\|^{1/n}$ then $\sum_{n=...
- 1,288
4
votes
1
answer
378
views
Ultralimits vs limits of subsequences
Let $(x_n)_{n \geq 1} \subset \mathbb{R}$ be a sequence of real numbers, and let $\omega \subset \mathcal{P}(\mathbb{N})$ be a non-principal ultrafilter. We say that $x$ is an ultralimit of $(x_n)_{n \...
- 3,085
0
votes
1
answer
44
views
for which $x$, is $f'(x)=lim_{n \to \infty}f_n'(x)$?
let $f_n(x)=\dfrac{x}{1+nx^4} \quad n=1,2... \quad$ and $f(x)=\lim_{n \to \infty}f_n(x)$
I have to determine for for which $x$, is $f'(x)=\lim_{n \to \infty}f_n'(x)$?
Here is what I have done:
$f_n'...
- 157
2
votes
0
answers
175
views
Decay rate of the coefficients in the Legendre series expansion
Let $u$ be a function in $L^2_{\nu}([0,\pi])$ where $\nu=2\sin(x)$
Let $\hat{u}_n$ denote the truncated Legendre series expansion of $u$ defined by
\begin{equation*}
\hat{u}_n:= \...
- 311
0
votes
2
answers
82
views
$\lim_{n \to \infty}\|x_{n}\|=\|x\|$ and and other conditions to imply $\lbrace x_{n} \rbrace \to x$.
Let $X$ be a metric space with inner product space. Suppose that there is a sequence, $ \lbrace x_{n} \rbrace $, in $X$ such $\lim_{n \to \infty}\|x_{n}\|=\|x\|$ and such $\lbrace \langle x_{n}, y \...
- 1,975
0
votes
0
answers
123
views
Linear limit extensions that are not Banach limits
By the Hahn-Banach theorem, we know that there is a (bounded) linear functional $\phi \in (\ell_\mathbb{R}^\infty(\mathbb N))^*$ extending the usual limit in the sense that it agrees with the limit on ...
- 1,477
1
vote
1
answer
78
views
$A$ relatively compact $\iff$ $A$ is bounded and limit of supremum of $A$ goes to $0$
Let $1\leq p < \infty$ and $A \subseteq \ell^{p}$. Show that:
$A$ relatively compact $\iff$ $A$ is bounded and $\lim\limits_{n \to \infty} \sup\limits_{x \in A}(\sum\limits_{i=n}^{\infty}\vert x_{...
- 1,838
10
votes
0
answers
753
views
Generalized limits
Cross-posted to Mathoverflow.
$\DeclareMathOperator{\Lim}{Lim}$
$\DeclareMathOperator{\dom}{dom}$
$\DeclareMathOperator{\shift}{\sigma}$
$\DeclareMathOperator{\cesaro}{C}$
After reading Terry Tao's ...
- 6,146
2
votes
1
answer
103
views
about application of egoroff’ theorem
$f_n:\mathbb{R}\rightarrow \mathbb{R}$ is measureable function ,for any x$\in \mathbb{R},\lim_{n\rightarrow \infty}f_n(x)=f(x)=1$,proof :there exist measureable set sequence {$E_k$},s.t. $f_n$ ...
- 453
-2
votes
2
answers
61
views
Find the limit as $n$ tends to infinity of $\sum_{k=1}^n \frac{n^{1/k}}{k}.$
Numerically, I think the answer converges to 2, but I couldn't come up with a function that it converges to pointwise. Also I couldn't think of a dominating function to apply to use Lebesgue's ...
- 181
1
vote
2
answers
556
views
Limit of Continuous Function of Convergent Sequence
Let $f:\mathbb{R}$ → $\mathbb{R}$ be a continuous function and ${a_n}$ be a convergent sequence in $\mathbb{R}$ with $\lim_{n\to\infty} {a_n} = a $ and $f(a) \neq 0$.
Show there there is a positive ...
- 119
4
votes
1
answer
614
views
The last digit of pi (in terms of Banach limits)
Let $\phi : l^\infty \to \mathbb C$ be a Banach limit, and define the sequence $\{x_k\}_{k\geq 0}$ to be the digits in the 10-base decimal expansion of $\pi$. Note that
$$\{x_k\}_{k\geq 0} \in l^\...
- 1,477
1
vote
2
answers
568
views
Necessary and sufficient condition for convergent series
Let $(a_i)_{i \in \mathbb{N}}$ be a sequence of positive reals such that
$$
\limsup_{i \rightarrow \infty} a_i \, i =0.
$$
Is this condition necessary and sufficient for $\sum\limits_{i=1}^\infty a_i &...
- 191
3
votes
1
answer
150
views
Functions with bounded derivatives, closed under composition
This is a follow-on of sorts to this question, but is self-contained.
Let $F_1 := \{f \in C^\infty(\mathbb{R}) \mid \|\frac{df}{dx}\|_\infty \le c_1\}$ ($c_i > 0$ throughout).
Given $f, g \in F_1$...
- 944
5
votes
3
answers
105
views
Prove "$ \lim_{n\to\infty}\|x_n\|_1 =0 \iff \lim_{n\to\infty}\|x_n\|_2 =0$" $\implies \|\cdot\|_1$ equivalent to $\|\cdot\|_2$
Let $X$ be a normed linear space and let $\|\cdot\|_i$ be arbitrary norms on $X$. Prove that the two arbitrary norms are equivalent if the following statement is true:
$$\lim_{n\to\infty}\|x_n\|_1 =0 ...
- 171
3
votes
2
answers
224
views
Does the sequence of operators $(A_nx)(t) = x(t^{1+\frac{1}{n}}) $converge by norm?
I have a simple question:
The question is as follows:
Consider the sequence of operators $A_n:C[0,1] \rightarrow C[0,1]$ as follows:
$$ (A_nx)(t) = x(t^{1+\frac{1}{n}})$$
Prove that for each $n\in \...
- 1,097
2
votes
1
answer
47
views
Question on sequence of $\;L^2\;$ integrable functions
Let $\;f_n\;$ be a sequence of functions such that $\;f_n \in
L^2((l_{-}^{f_n},l_{+}^{f_n}),\mathbb R^m)\;$ where $\;-\infty\le
l_{-}^{f_n} \lt l_{+}^{f_n} \le +\infty\;$. If I knew $\;\lim_{n \to
...
- 2,974
0
votes
1
answer
38
views
Double limit $f_n(t_n) \to 0$ implies something about the limit of $f_n$?
Suppose for each $n$, $f_n\colon (0,T) \to H$ is a map where $H$ is a Hilbert space. We know that
$$f_n(t) \to f(t)\quad(n \to \infty)$$ for fixed $t$ and $$f_n(t) \to 0\quad(t \to 0)$$
for fixed $n$ ...
- 475
2
votes
1
answer
65
views
Keep getting wrong answer for showing $(a_n)=(\frac{2n^2-1}{n^2},\frac{1}{n})$ is $d^{(2)}$-convergent?
A sequence in $\mathbb{R}^2$ is given by:
$a_n=(\frac{2n^2-1}{n^2},\frac{1}{n})$, for each $n∈\mathbb{N}$.
I must show that $(a_n)$ is $d^{(2)}$-convergent.
My textbook says that I can show $a_n$ ...
- 387
0
votes
1
answer
188
views
Application of Cauchy condition for uniform convergence
There is a sequence of functions $f_n(x)=\frac{x}{n}$. It converges pointwise on $R$ to $0$ (here $x$ can't be equal to $\infty$ because $\infty$ isn't in $R$):
$$
\lim_{n\rightarrow \infty} f_n(x)=\...
- 2,113
1
vote
0
answers
84
views
Is this a usual approach to uniform convergence?
Consider $E\subset \mathbb{R}$ and a sequence of functions $(f_n)_{n\in \mathbb{N}}$ with $f_n : E\to \mathbb{R}$. The easiest form of convergence we can define is the pointwise convergence - we say ...
- 27.2k
1
vote
1
answer
29
views
Convergence of a sequence in the sense of $\mathfrak{L}^{\text{loc}}_{1}(\mathbb{R},\mathbb{R})$
Define $w_{n}\in \mathfrak{L}^{\text{loc}}_{1}(\mathbb{R},\mathbb{R})$ such that
$$w_{n}(t)=\begin{cases}0, & |t|<n, \\ n, & |t|\ge n.\end{cases}$$
I want to prove that $w_{k}$ converges ...
- 1,068
5
votes
1
answer
1k
views
Prove a sequentially compact metric space is bounded.
Prove that if the metric space $(X, d)$ is sequentially compact, that there exists points $x_0$ and $y_0$ belonging to $X$ such that;
$$d(x, y) \leq d(x_0, y_0)$$ for every $x$ and $y$ belonging to $...
- 157
0
votes
2
answers
112
views
prove that $\lim_{n \to \infty} \lambda x_{n} = \lambda \cdot L $ for any $\lambda \in \mathbb{R}$ and $\lambda \neq 0$
I asked this question since I have already proven the easy case for when $ \lambda = 0$
prove that $ lim_{n \to \infty} \lambda x_{n} = \lambda \cdot L $ for any $\lambda \in \mathbb{R}$ and $\...
- 59
3
votes
1
answer
93
views
Limit of a functional
I'd like to find:
$$
\lim_{\varepsilon\rightarrow 0}\frac{\varepsilon}{\varepsilon^2+x^2}\qquad \mbox{ in }\mathcal D'(\mathbb{R})
$$
And I started with the definition:
$$
\left\langle \frac{\...
- 395
3
votes
0
answers
430
views
Passing to the limit in a PDE; problem with subsequence (please check my answer)
For $w \in L^2(0,T;H^1)$, consider the PDE
$$\int u'(t)v(t) + \int g(w(t))\nabla u(t) \nabla v(t) = \int f(t) v(t)\quad \forall v \in L^2(0,T;H^1)$$
where $u \in H^1(0,T;L^2)\cap L^2(0,T;H^1)$, and $f$...
- 171
1
vote
1
answer
199
views
Can I conclude that $F\in\ell^1$?
Let $\ell^\infty$ be the Banach space of real bounded sequences with its usual norm and $S\subset\ell^\infty$ be the space of convergent sequences. Define $f:S\rightarrow\mathbb{R}$ by $$f(x)=\lim_{n\...
- 22.9k
4
votes
2
answers
277
views
Limit at infinity
I really want help in this problem: given a sequence of pairs $(x,y)$ in the $xy$-plane
$$S=\left\{\left(n, \frac{-1}{\sqrt{n}}\right)\right\}_{n=1}^{\infty}\;,$$
how to find
$$\lim_{x\to \infty} \...
- 41