# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 $$\sum_{n=1}^{\infty} \|x_n\| < \infty \wedge \sum_{n=1}^{\infty} x_n \notin X,$$ ...
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### 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 ...
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### 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 ...
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### 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 $(\mathbb{R},\left\| \cdot \right\|)$.
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### 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 ...
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### 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 ...
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### 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 $E$ ...
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### 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 ...
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### 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?
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### 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 ...
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### 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, ...
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 ...