3
votes
1answer
38 views

Convergence of $a_n=d(u_n,\Bbb{Z})$ where $u_n=\bigl(\frac{1+\sqrt{5}}{2}\bigr)^n$.

I would like to study the sequence defined by $$ a_n=d(u_n,\Bbb{Z}). $$ Where $u_n=\bigl(\frac{1+\sqrt{5}}{2}\bigr)^n$ and $d(u_n,\Bbb{Z})=\inf\{d(u_n,x):x\in\Bbb{Z}\}$. I do not really ...
1
vote
1answer
43 views

Why does this proof fail?[convergence of infinite sums]

An equivalent way of saying that a normed vector space is complete is saying that every absolutely convergent series, converges. Hence' in some normed vector-space(incomplete), there must be a ...
2
votes
1answer
53 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 $$\sum_{n=1}^{\infty} \|x_n\| < \infty \wedge \sum_{n=1}^{\infty} x_n \notin X,$$ ...
2
votes
0answers
34 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 ...
0
votes
0answers
71 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 ...
0
votes
1answer
59 views

Show that the sequence converges to 0 under any norm in the space (R,‖.‖) [closed]

Show that the sequence $a_n = 1/n^2$ converges to 0 under any norm in the space $(\mathbb{R},\left\| \cdot \right\|)$.
1
vote
2answers
120 views

If a sequence of summable sequences converges to a sequence, then that sequence is summable.

Let $(a_i)^n$ be a sequence of complex sequences each of which are summable (they converge). Then if they have a limit, the limit sequence $(b_i)$ is also summable. All under the sup norm for ...
0
votes
1answer
309 views

Is this valid: Every Cauchy sequence in a normed space is absolutely convergent.

Proof. Let $X$ be a normed space with norm $|\cdot |$ and $(x_n)$ be Cauchy. Then for all $\epsilon \gt 0, \ \exists N : m,n \gt N \implies |x_m - x_n| \lt \epsilon$ is the standard definition of ...
3
votes
2answers
22 views

Density and the size of coefficients

Let $E$ be a Banach space, and $F$ a dense subspace spanned by a countable base $y_i$ of unit norm. Let $x \in E$ and $x_n = \sum_{i_n=1}^{N_n} a_{i_n} y_{i_n}$ be a sequence of elements of $E$ ...
4
votes
2answers
146 views

Why doesn't the Weierstrass approximation theorem imply that every continuous function can be written as a power series?

I hope that my question in the title is well formulated. I am a little bit confused with the next exercise from a book: Argue that there exists functions $f \in C[0, \frac{1}{2}]$ that cannot be ...
5
votes
1answer
159 views

Is $(l^1 ,\|.\|)$ a Banach space?

Suppose $x=\{x_n\}\in l^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$, let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $l^1$ . Is $(l^1 ,\|.\|)$ a Banach space?
1
vote
1answer
100 views

Proving $\ell_\infty$ is complete

I start by taking a Cauchy sequence $(a_i)$ in $\ell_\infty$. I denote the terms of $(a_i)$ as $f_1, f_2, f_3, \dots$ and so on. For each $x \in \mathbb{N}$, the sequence $(f_1(x), f_2(x), f_3(x), ...
5
votes
3answers
284 views

If $\{x_n\}$ is a Cauchy sequence in a normed vector space, is $\frac{x_n}{\|x_n\|}$ Cauchy?

Let $\{x_n\}$ a Cauchy sequence in a normed vector space $X$. Is $$y_n = \frac{x_n}{\|x_n\|}$$ another Cauchy sequence in $D = \{x\in X : \|x\| = 1\}$? Remark: The idea is prove that if $D$ is ...
1
vote
1answer
33 views

closed subspace of $\ell_1$ such that sequences of alternating terms cover $\ell_1$

If $X$ is a closed subspace of $\ell_1$ such that every sequence $y=(x_{2n})\in\ell_1$ can be seen as the 'every other term' sequence given by some $x=(x_n)\in X$, why must there be a constant $C$ ...
3
votes
1answer
165 views

convergence of sequence of averages the other way arround

In a vector normed space, if $ \{x_n\} \longrightarrow x $ then $ z_n = \frac{x_1 + ...+x_n}{n} \longrightarrow x $ Is it true the other way arround too? meaning: if $ z_n = \frac{x_1 + ...
3
votes
2answers
165 views

convergence of weighted average. proof [duplicate]

It is well known that for any sequence $\{x_n\}$ in a normed space which converges to a limit $x$, the sequence of averages of the first $n$ terms is also convergent to $x$. That is, the sequence ...
3
votes
1answer
1k 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
1answer
66 views

Something weaker than the Riesz basis

I have some function $f$, real valued and continuous. I formed functions $\{f_{m,k}, k \in \mathbb{Z}, m>0\}$ such that that $\mathrm{span}\{f_{m,k}, k \in \mathbb{Z}, m>0\}$ is dense in ...
5
votes
2answers
338 views

If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $

I have this problem I'm working on. Hints are much appreciated (I don't want complete proof): In a normed vector space, if $ x_n \longrightarrow x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} ...
12
votes
5answers
287 views

Passing from induction to $\infty$

Somehow, the operation of passing to the limit after I have shown that something is true by induction for each natural number $n$ troubles me each time. I know there are instances where one cannot ...