1
vote
1answer
30 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
29 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
20 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
106 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
34 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
60 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
25 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
48 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
46 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 ...
1
vote
1answer
32 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
0answers
32 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 $\displaystyle\sum_{q=0}^{\infty} (M_q-\beta_q)t^q=(1+t)Q(t)$ ...
2
votes
0answers
53 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
25 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
35 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
39 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
54 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
22 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
74 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
97 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
166 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 ...
1
vote
0answers
41 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 ...
0
votes
1answer
28 views

Bessel Sequence proof check.

I have a similar question to definition of Bessel sequence, where it was solved using Banach-Steinhaus. A sequence $\{f_k\}_{k=1}^{\infty}$ is called a Bessel sequence in a Hilbert space ...
3
votes
3answers
45 views

Pointwise Convergence in $L^1$ norm

Suppose we have a sequence of functions $\left\{f_n\right\}_{n=1}^\infty\subseteq C^1([0,1])$ and $f_n\to f\in C([0,1])$ in the $L^1$ norm and $f_n'\to g\in C([0,1])$ in $L^1$. Does it follow that ...
1
vote
0answers
50 views

Sequence of $r_n:=\frac{x_{n+1}}{x_n}$ where $x_n$ is a sequence

Let $f:X\rightarrow X$ be a function on some complete metric space $(X,d)$. Let, $x_n$ is a sequence in the metric space defined by $x_{n+1}=f(x_n)$ and starting from $x_0$. My questions are (1) ...
2
votes
1answer
49 views

definition of Bessel sequence

A sequence $\{f_{k}\}_{k=1}^{\infty}$ is called a Bessel sequence in a Hilbert space $H$, if there exists $B>0$ such that $$\sum_{k=1}^{\infty}|\langle f,f_{k}\rangle|^{2}\leq B\|f\|^{2}$$ for all ...
5
votes
2answers
59 views

Do non-$\ell^2$ sequences have an $\ell^2$ functional that takes them to infinity?

Suppose $\{a_n\}_{n=1}^{\infty}$ is a sequence of real numbers (suppose also positive for simplicity) so that $$\sum_{n=0}^{\infty} a_n^2 = \infty$$ i.e. the sum diverges. Can you necessarily find a ...
0
votes
0answers
17 views

How to factorize exponential as a convolution of finite number of functions( series)?

We choose a points $x_{1},..., x_{n}$ in $\mathbb Z$ (not 0), such that $$x_{k+1}\not = x_{1}+x_{2}+...+x_{k} \ \ (k=1,2,...,n-1).$$ Let $\delta_{x}$ denote the measure of total mass $1$, ...
0
votes
1answer
18 views

Pitt's theorem on automatic compactness of bounded operators between sequence spaces

Why is it called Pitt's theorem? I couldn't locate the origin of the statement.
3
votes
2answers
59 views

Find a sequence of positive functions with non-trivial properties in $L^1([-\pi,\pi])$ and in $L^2([-\pi,\pi])$

I was asked to exhibit a sequence of positive functions $\{f_n\}_{n\in\mathbb{N}}$ belonging to $L^2([-\pi,\pi])$ such that: $\{f_n\}_{n\in\mathbb{N}}$ is strongly converging to $0$ in ...
2
votes
1answer
107 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 ...
2
votes
0answers
78 views

Troublesome proof in Functional Analysis with dual vector space

Greetings to all of you I have tried to prove the following theorem but I am having some troubles with it. Let $X$ be a separable normed space and $(x_n')$ a bounded sequence in $X'$, then there is a ...
1
vote
1answer
25 views

$\|(I+A)^{-1}\| \leq \frac{1}{1-\|A\|)}$

I have the following problem, of which I have a slight problem to finish with the second part: Let $X$ be a Banach space and let $A \in B(X)$, $\|A\| < 1$. Prove that $(I+A)^{-1}$ exists and is ...
2
votes
3answers
58 views

Are the convergent sequences dense in the bounded sequences?

Since it would be comfortable for something I am currently trying to prove if this would hold I wanted to ask here whether it is true that $c$ is dense in $l^{\infty}(\mathbb{N})$?
1
vote
1answer
44 views

Redundance in $l^p$ space.

I am covering the following problem: Let $A \subset l^p$, with $p \in [1,\infty)$, then is equivalent: i)A is relatively compact ii)A is bounded and we have $$\lim_{n \rightarrow \infty} \sup_{x ...
1
vote
1answer
91 views

Prove that the sequence of L-Lipschitz functions converge

$f_n(x): [a,b] \to \mathbb R$ are a sequence of functions that all are $L$-Lipschitz: means - $|f_n(x)-f_n(y)| \le L|x-y|$ , ($L$ is for all the functions) and assume $f_n \to f$ in a pointwise ...
0
votes
1answer
63 views

Constructing Sequences in Lp

Consider Banach Space $L^{p}(U)$ where $U$ is bounded open subset in $\mathbb{R}^{n}$. Take a bounded sequence $\{u_{m}\}_{m}^{\infty}$ in $L^{p}(U)$. Consider a subsequence ...
0
votes
1answer
53 views

Sequence in span with disjoints supports has (?) block subsequence.

Assume $b_k \in <e_i, \text{with coefficients} \ a_i^k \geq 0>$ is a sequence and $a_i^k$ have disjoint supports(support is the set where $a_i^k \neq 0$). Is there a way to prove or disprove ...
2
votes
1answer
67 views

Additional hypotheses needed for the converse of $\mathscr F \subset H(G)$ normal $\implies$ $\mathscr F ' = \{f' : f \in \mathscr F\}$ normal?

Problem Show that if $\mathscr F \subset H(G)$ is normal then $\mathscr F' = \{f' : f \in \mathscr F\}$ is also normal. Is the converse true? Can you add something to the hypothesis that $\mathscr ...
0
votes
0answers
63 views

Help for applying Hahn Banach theorem in this exercise .

I want to use this version of Hahn Banach: $X$ real vector space, $M$ a subspace, $p$ a sub-linear mapping and $T:M\rightarrow\mathbb{R}$ linear and $T(x)\leq p(x)$ $\forall x \in M$ Then ...
2
votes
0answers
87 views

Is there a “natural” norm in the space of all sequences?

Let $X$ be $\mathbb{R}^\mathbb{N}$, the space of all real-valued sequences. Is there a natural norm in this space?
0
votes
1answer
26 views

Holder Space series term

Holder space $C^{k,\gamma}(\bar{U})$ has the norm $||u||_{C^{k,\gamma}(\bar{U})} := \sum_{|\gamma | \leq k}||D^{\alpha}u||_{C(\bar{U})} + \sum_{|\gamma|=k}[D^{\alpha}u]_{C^{0,\gamma}(\bar{U})}$, where ...
2
votes
1answer
98 views

Prove that the Fourier series of $\dfrac{1}{f}$ is absolutely convergent.

I have a problem: Let $f$ be a continuous function on the unit circle $(\Gamma)$: $$\Gamma=\{e^{i\theta}: \theta\in [0, 2 \pi]\}$$ Assume that $f \ne 0$ on $\Gamma$, and the Fourier ...
4
votes
0answers
96 views

Showing a sequence $x_{n+1} = Tx_n$ forms a contraction mapping

I want to show that the sequence given by $$x_{n+1} = Tx_n = x_n-\frac{(x_n^2-2)}{x_n+x_{n-1}}$$ forms a contraction mapping. That is $$|Tx_1-Tx_2|\leq c|x_1-x_2|.$$ Where $c$ is to be determined. I ...
0
votes
1answer
25 views

function on a fixed length sequence of positive real number that induces lexicographic order

Let $S$ be the finite set of sequences of length $n$, whose entries are all real positive numbers. Can we define a function $f$ on $S$ such that the order $f$ induces on $S$ i.e $\le$ is the same as ...
1
vote
0answers
66 views

Sequences of 'Rayleigh-like quotients' and their minima for a symmetric positive semi-definite matrix

Let $A$ be an $N\times N$ symmetric positive semi-definite matrix with eigenvalues $0 \leq \lambda_1 \leq \lambda_2 \leq \dots \leq \lambda_N$ and corresponding eigenvectors $u_1, u_2, \dots, u_N$. ...
0
votes
2answers
129 views

The Fourier series converges absolutely $\implies$ it converges uniformly.

Let $S_N(f)$ be the $N$th partial sum of the Fourier series for $f$. I.e. $$S_N(f) = \sum_{n = -N}^{N} \hat{f}(n) e^{2\pi i n x / L}$$ Suppose that the Fourier series converges absolutely, i.e. ...
2
votes
1answer
76 views

Is there a generalised parallelogram inequality in uniformly convex spaces

A version of generalized parallelogram inequality in Hilbert spaces is $\sum_{i=1}^{n}\|\alpha_i x_i\|^2=\sum_{i=1}^{n}\alpha_i\| x_i\|^2-\sum_{i\neq j}^{n}\alpha_i\alpha_j\| x_i-x_j\|^2$ ...
3
votes
2answers
81 views

Determining if a sequence of functions is a Cauchy sequence?

Show that the space $C([a,b])$ equipped with the $L^1$-norm $||\cdot||_1$ defined by $$ ||f||_1 = \int_a^b|f(x)|dx ,$$ is incomplete. I was given a counter example to disprove the statement: Let ...