# Questions tagged [schauder-basis]

A Schauder basis is a basis that use linear combinations that may be infinite sums..

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### basic sequence is equivalent to the unit vector basis of $C_0$

I study from "ِ sequences and series in Banach spaces" by j.Diestel A series $\sum_n x_n$ is said to be weakly unconditionally Cauchy (wuC) if, given any permutation $\pi$ of the natural ...
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### Are $(x_n-x_{n-1})$ and $(x_1+...+x_n)$ Schauder basis?

My friend and I were talking about challenging problems as we prepare for our finals. She suggested three intriguing ones that caught my interest, and I thought of sharing my solutions to them. Feel ...
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### Example of a point that is not the limit of any sequence in a connected topological space

Question: Let $X$ be a connected space with a topology not necessarily sequential. What is an example where a point in $X$ is not the limit of any not eventually constant sequence? Motivation. ...
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### Counting, tracking/organizing Schauder Basis for Infinite-Dimensional Spaces?

I'm thinking of spaces like $$L^2[a,b] ; -\infty <a<b < \infty$$, and Schauder bases, such as that given by $$\{\pi, Cos(n\pi), Sin(n \pi); n=1,2,...\}$$ If $$V/F$$ is a finite dimensional ...
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### In a Hilbert space $\mathbb{H}$, find a complete and linearly independent sequence $(x_k)$ which is not a Schauder basis.

this is one of the problems I found in a book I'm studying: In a Hilbert space $\mathbb{H}$, find a complete and linearly independent sequence $(x_k)$ which is not a Schauder basis. With it, there is ...
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### Banach space with unconditional basis and its subspaces

I am interseted in the following problem. By Bessaga and Pelczynski paper On bases and unconditional convergence of series in Banach spaces we know that if $X$ is a Banach space with a basis $(x_n)$ ...
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### Faber-Schauder basis of $C[0, 1]$

This question may be worded a bit vaguely, but I would like to understand the relevance of the notion of Schauder basis. This notion seems to be of particular interest to Banach space theorists, but ...
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### summing basis of $c_0$ is a conditional basis

The standard unit vector basis $(e_n)_{n=1}^\infty$ is an unconditional basis of $c_0$ and $l^p$ for $1 \leq p < \infty.$ An example of a Schauder basis that is normalized conditional (i.e., not ...
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### 12 Possible definitions of basis and their properties

Let $I$ be any non-empty set, one can consider the set $2^I_{fin} := \{ J \in 2^I \; : \; J \text{ is finite } \}$ So basically $2^I_{fin}$ consists of all the finite subset of $I$. Then the ordered ...
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### Weak p-summability of Schauder basis

It is know that in a Hilbert space $H$, an orthonormal basis $(e_i)_{i\in \mathbb{N}}$ is weakly 2-summable, that is, $\sup_{h\in B_H}\sum_{i=1}^\infty \langle h,e_i\rangle^2<\infty$. I was ...
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### Closed subspaces of separable Banach spaces with Schauder basis need not themselves have a Schauder basis?

Let $B$ be a Banach space with a Schauder basis (thus $B$ is separable), and let $X \subseteq B$ be a closed subspace of $B$ (thus a separable Banach space itself). Claim: $X$ need not have any ...
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### Standard basis forms a Schauder basis for $\ell^p, p\in [1,\infty)$.

Standard basis, $(e_n)_{n=1}^{\infty}$ in which $e_n=(\delta_{k,n})_{k=1}^{\infty}$ forms a Schauder basis for $\ell^p, p\in [1,\infty)$. My proof is as follows: It is obvious that $e_n \in \ell^p$ ...
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### Prove $\sup_{1 \leq n < \infty} \|\sum_{i=1}^n \alpha_i x_i\| \leq \lim_{n \rightarrow \infty} \sup_{1 \leq k \leq n} \|\sum_{i=1}^k \alpha_i x_i\|$

From book bases in Banach spaces I by ivan singer Proposition Let $\{x_n\}$ be a sequence in a Banach space $E,$ such that $x_n \neq 0 (n = 1,2,...),$ and let $Y$ be the Banach space of sequences of ...
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