2
votes
1answer
49 views

Series in a space which is not complete

Let $X$ be a normed vector space and $\left\lbrace x_n \right\rbrace_{n \in \mathbb{N}} \in X^{\mathbb{N}}$ with $$\sum_{n=1}^{\infty} \|x_n\| < \infty \wedge \sum_{n=1}^{\infty} x_n \notin X,$$ ...
1
vote
0answers
29 views

Weakly closed convex set and norm convergent

Let $X$ be a normed space and let $\{x_n\}$ be a sequence in $X$ such that $x_n\to x$ weakly. Show that there is a sequence $\{y_n\}$ such that $y_n\in \operatorname{co}\{x_1,...,x_n\}$ and ...
1
vote
1answer
46 views

Need help considering series like these: $\sum_{n=1}^\infty\langle x,e_n\rangle e_n$

I'm working in a Hilbert space $H$ with ONB $(e_n)$ and I have $\alpha=(\alpha_n)\in\ell^\infty$. I have an operator that looks like this: $$T_\alpha x=\sum_{n=1}^{\infty}\alpha_n\langle ...
0
votes
1answer
16 views

Multiplication of non-summable sequences convergent to $0$

Let $(\lambda_n)_n$ be a sequence of real numbers which converges to $0$ (i.e., is in $c_0$), but is not in $\ell^p$ for any $1\leq p<\infty$, e.g., $\lambda_n=\frac{1}{\log(n+2)}$ for ...
4
votes
1answer
94 views

On an estimate of sequences with weights

Does there exist a $C > 0$ such that $$ \sum_{n \geq 1} a_n \leq C \left( \sum_{n \geq 1} 2^n a_n^2 \right)^{1/4} \left( \sum_{n \geq 1} 2^{-n} a_n^2 \right)^{1/4} $$ for all $a_n \geq 0$ with ...
0
votes
0answers
52 views

a completion of $c_{00}$ under some norm

Let $X$ be the completion of $c_{00}$ under some norm $\|\cdot\|_X$ satisfying $\|e_n\|_X=1$ for all $n$, where we let $(e_n)$ denote the canonical unit vectors in $c_{00}$. In Albiac/Kalton's book ...
4
votes
0answers
42 views

a question about Tsirelson's space

Background. Let $T$ denote the Figiel-Johnson construction of the Tsirelson space, that is, the completion of $c_{00}$ under the implicitly-defined norm ...
1
vote
1answer
33 views

How implies “$\sum_{k=1}^\infty z_nx_n$ exists for every x $\in c_0$” that z $\in l^1$?

If z is a sequence in $\mathbb R$ and $\sum_{k=1}^\infty z_nx_n$ exists for every x $\in c_0$, how follows that z must be in $l^1$?
3
votes
1answer
16 views

Subspace of certain series in a Hilbert space is compact

Let $E$ be a Hilbert space and let $\{x_{n}\}$ be an orthonormal basis.  Let $\{c_{n}\}$ be a sequence of positive numbers such that $\sum c_{n}^{2}$ converges.  Let $C$ be the subset of $E$ ...
1
vote
1answer
48 views

What is wrong with this “counterexample” of boundedness of weakly convergent sequences?

Weakly convergence sequences $\{u_n\}$ in a Hilbert space $H$ are bounded. Here is an attempted "counterexample". What is wrong with this? Let $H = \ell_2(\mathbb{N})$, and let $\{e_n\}$ be the ...
2
votes
1answer
26 views

strengthen the condition of convergence in measure of sequence of functions

Let $\{f_n\}_{n=1}^\infty$ be a sequence of non-negative measurable functions on a measurable set $E$. (1). Suppose for any $\epsilon>0$, $$\sum_{n=1}^\infty \mu\{x\in E: ...
1
vote
1answer
46 views

I'm searching for the formula of the series $ \sum_{n=0}^{\infty}a^{n^l} $

I'm searching for the sum-formula (if exists) of the following power series: $$ \sum_{n=0}^{\infty}a^{n^l} $$ where $l=2,3,....$, and $|a|<1$.
1
vote
1answer
26 views

Convergence of $\chi_{\{x \mid u_n(x) \in I\}} \to \chi_{\{x \mid u(x) \in I\}}$ for $u_n \to u$?

Let $u_n \to u$ in $L^2(\Omega)$ and let $I$ be an bounded interval. Does it follow that $$\chi_{\{x \mid u_n(x) \in I\}} \to \chi_{\{x \mid u(x) \in I\}}$$ at least for a subsequence of $u_{n_j}$ ...
1
vote
1answer
22 views

conditions for Convergence of sequence of functions

Suppose $\{f_n\}$, $n \in \mathbb{N}$ is a sequence of a positive real-valued functions defined on $[0, T]$ and continuous on $(0, T)$. If {$f_n$} satisfies the following conditions : $f_n( iT/2^n ) ...
1
vote
1answer
72 views

Show that $\limsup_{n\rightarrow\infty} \|x_n\|^{\frac{1}{n}}=1$

Let $(x_n)$ be weakly convergent, but not norm convergent, sequence from a Banach space. Show that $\limsup_{n\rightarrow\infty} \|x_n\|^{\frac{1}{n}}=1$. Any help?
2
votes
1answer
23 views

Denseness: $\overline{\mathcal{l}^2_0}=\mathcal{l}^2$

How to prove that the finite sequences are indeed dense within the space?
1
vote
1answer
41 views

Small question about strong convergence

I have a small question I have that $ \lambda_1 ||v_0||^2_{L^2(0,1)}=||v_0||^2_{H^1_0(0,1)}$ and that $v_n\rightarrow v_0$ on $L^2$ (strongly) From the Poicaré inequality i have that ...
1
vote
1answer
36 views

Convergence $ \ell^1$ sequences

if I have a sequence $(x_n) \in \ell^1$ and an element $x \in \ell^1$ and we have that for all $k \in \mathbb{N}: x_n(k) \rightarrow x(k)$, does this mean that $||x_n-x||_1 \rightarrow 0$?
3
votes
1answer
82 views

Rudin's 'Principle of Mathematical Analysis' Problem 7.12

Suppose $g$ and $f_n$ ($n = 1,2,\ldots$) are defined on $(0,\infty)$, are Riemann-integrable on $[t,T]$ whenever $0 < t < T < \infty$, $|f_n| \leq g$, $f_n \rightarrow f$ uniformly on every ...
4
votes
1answer
87 views

Euler's Refutation of Fermat's Conjecture

Fermat postulated that all numbers of the form $$2^{2^n}+1$$ are prime (where n = any integer). Then Euler came along with a rather ingenious proof that this was not, in fact the case. I came across ...
2
votes
1answer
55 views

the principle of uniform boundedness and $l^p$ space

If $1<p<\infty$ and $\{x_n\}\subset l^p$, then $\sum_{j=1}^\infty x_n(j)y(j)\to 0$ for every $y\in l^q$, $\frac{1}{p}+\frac{1}{q}=1$, iff $\sup_n||x_n||_P<\infty$ and $x_n(j)\to 0$ for every ...
0
votes
1answer
18 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 ...
0
votes
1answer
39 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) ...
1
vote
2answers
66 views

Absolute convergence does not implies convergence

Find a space and a series that converges absolutely but it does not converges. It is clear that the space can't complete or Banach.
1
vote
1answer
43 views

family of functions/sequences taken over reals instead of naturals

How does convergence for a sequence and a family of functions change when considering $n$ taken from $\mathbb{R}$ instead of the $\mathbb{N}$? for example, consider mollifiers which are defined as ...
4
votes
1answer
40 views

Sequence spaces

Suppose the sequence spaces $$d \colon=\left\lbrace \left\lbrace x_n\right\rbrace_{n \in \mathbb{N}} \in \mathbb{K}^{\mathbb{N}} \colon x_n=0 \ \text{for almost all} \ n \right\rbrace$$ and ...
0
votes
0answers
26 views

Convex Feasibility problem on Hilbert space

Let $H$ be a Hilbert space with real inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$. Consider a closed convex set $X \subset H$ and let $P_X: H \rightarrow X$ be the ...
5
votes
1answer
62 views

Which properties do still hold for the limit of a sequence of functions?

I think it would be very useful to have a list of properties that are preserved for the limit of a sequence of functions, and was wondering if you could help me making the list more complete. Let ...
1
vote
1answer
35 views

prove $\lim_{n\to\infty}\sup\{|f(x) - f_n(x) | : x \in S \} =0$

I understand how the theorem works but how would you prove that a sequence $f_n$ of functions on set $S \subset \mathbb{R}$ converges uniformly iff $$\lim_{n\to\infty} \sup\{|f(x) - f_n(x) | : x \in ...
0
votes
1answer
37 views

German and French word for Basic sequence

Wikipedia says that a sequence $(x_n)$ is a basic sequence iff it is a Schauder basis of its closed linear span. I was wondering whether there is a French or German word for 'basic sequence'? How ...
0
votes
1answer
29 views

Fixed Point Iterations for Bounded Affine Functions

Let $f: X \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ compact and convex. From Brouwer fixed-point theorem, $f$ admits a fixed point ($\bar{x} \in X$ such that $f(\bar{x}) = \bar{x}$). ...
1
vote
0answers
110 views

Is this sequence a Cauchy sequence?

Consider a continuous mapping $f: X \rightarrow X$, where $X \subset \mathbb{R}^n$ is compact and convex. Consider a sequence $\{x_k \in X\}_{k \geq 0}$ such that for all $k \geq 0$ and $h \geq 1$, ...
0
votes
1answer
43 views

Convergence of series of continuous bounded functions

Let $\mathcal{C}$ be the set of continuous bounded mappings from $\mathbb{R}^n$ to $\mathbb{R}^m$. Let $F: \mathcal{C} \rightarrow \mathcal{C}$ be a continuous (with respect to the $\sup$ norm) ...
0
votes
1answer
61 views

Is this construction already a contradiction

Let's say we have $\ell^p$ for $p>2$ and $\ell^2 \subsetneq \ell^p$. Let $U$ be a closed subspace of $\ell^2$. Is it possible that $U$ is isomorphic to $\ell^p$ as a Banach space?- I hardly think ...
0
votes
1answer
32 views

show that $(x_k)$ is convergent and limit$\notin\ell^p$

Let $\ell^p:=\{(x_n)|\|x_n\|_{\ell^p}<\infty\}$ be the space of sequences with finite $\ell^p$-norm. Show that the sequence $x_1:=1$, $x_k:=\frac{1}{\log k}$, $k\geq 2$ converges to $x=0$, but ...
0
votes
2answers
50 views

$l^r \subset l^p$ and is it even a subspace

It is true that for $r<p$ and $r,p \in [1,\infty)$ we have that $l^r \subset l^p$. Is it true that $l^r$ cannot be isomorphic to a subspace of $l^p$?
2
votes
1answer
55 views

I have to decide if $ \ell^1\subset c_0$ is closed or not.

I was asked to decide if $ \ell^1\subset c_0$ is closed or not, where $$\ell^1=\{(x_n)_{n\in\mathbb N}\subset\mathbb R:\sum_{n=0}^{\infty}|x_n|<\infty\}$$ $$c_0=\{ (x_n)_{n\in\mathbb ...
2
votes
1answer
44 views

$(\ell_p,\|.\|_{\infty})$ Banach or separable

Is $(\ell_p,\|.\|_{\infty})$ for $1\leq p<\infty$ a Banach or separable space? is there any fast way of proving it without checking separability or completeness with the usual way?
0
votes
1answer
39 views

Question on Morse inequalities

I want to understand why: if i have then $(4.1)$ is formal : it means that please help me Thank you EDIT1: $(4.1)$ tel us that $\displaystyle\sum_{q=0}^{\infty} (M_q-\beta_q)t^q=(1+t)Q(t)$ ...
4
votes
0answers
79 views

Is this Neumann series solution unique?

I have a Fredholm integral equation of the second kind given as $$f(x)=g(x)+\lambda\int_{-\infty}^\infty K(x,y)f(y)dy, $$ where $\lambda\in(0,1)$, the kernel $K(x,y)=\phi(x-y)$ is a Gaussian ...
1
vote
0answers
30 views

Contraction Mapping Conditions

Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ continuous and uniformly bounded, and consider the iteration $$ x^{(k+1)} = \frac{1}{k} \sum_{i=1}^{k} f\left( x^{(i)} \right) + x^{(i)} $$ for ...
1
vote
1answer
39 views

What is the main defferences between nets and ordinary sequences

I know that there are many results in metric spaces (or first-countable topological spaces) can be describe in the language of sequences but these results might not be true in general topological ...
3
votes
1answer
50 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 ...
0
votes
1answer
21 views

can a finite sequence of real numbers in range given other sequenes of real numbers in domain be fit by several functions?

So let us say that there is time variable $t$ that can only be natural number. And then, for each $t$ there are data for each variable $a$, $b$, $c$, and so on. And then we have variable $y$. We want ...
0
votes
1answer
19 views

Density of series the functions

Let $(g_n)_{n\in\mathbb{N}}\subseteq {\cal C}^0([a,b],\mathbb{R})$ with $\sum\int_a^b|g_n(t)|dt < \infty$ Prove that $\sum|g_n(t)|$ converges Show that ...
3
votes
0answers
72 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 ...
1
vote
1answer
37 views

Check continuity and find norm of linear functional

I have to check continuity and find norm of the following linear functional $$\ell^{2}\ni \{x_n\}^{\infty}_{n=0} \rightarrow \sum^\infty_{n=0}\frac{x_n}{\sqrt{2^n}}\in \mathbb{K}$$ As for continuity, ...
3
votes
1answer
83 views

Fourier series to calculate infinite series

I try to show that $\sum_{i=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ using Fourier series and $f(x) = x$ on $L^2_{\mathbb{C}}[-\pi, \pi]$, with basis $e_n(x) = \frac{1}{\sqrt{2\pi}}e^{inx}$. I ...
0
votes
2answers
131 views

Is the sequence of function $\frac{\sin nx}{\pi x}$ convergent to $\delta(x)$

I have no doubt that the integral of the function from negative infinity to positive infinity is 1. But I don't think $f_n(x) = \frac{\sin nx}{\pi x} \rightarrow 0 $ as $ n \rightarrow \infty $ given ...
2
votes
3answers
183 views

Continuity of Fixed Point

For all $a \in \mathbb{R}$, let $f_a: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and contractive, that is, there exists $\epsilon \in (0,1)$ such that $\left\| f_a(x)-f_a(y) \right\| \leq ...