# Questions tagged [schauder-basis]

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

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### Schauder basis $\implies$ Separable for non translation invariant metric linear spaces

It is fairly straightforward to prove that over a normed space $(V,\| \cdot \|)$ the existence of a Schauder basis $\{ e_n\}_{n=1}^\infty$ implies the separability of the space. I was however ...
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### Cardinality of a Banach space

Let $X$ be a separable infinite-dimensional Banach space. Assume that every element of $B_{X^{**}}$ is the weak-star limit of a sequence in $B_{X}$. Clearly, by the canonical embedding, we can see ...
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### Prove that a Schauder basis consists of isolated points.

Let $X$ be a Banach space. Suppose $(x_n) \in X$ is a sequence such that every $x \in X$ has a unique representation in the form $x = \sum_{n = 1}^{\infty} \lambda_ n x_n$. Prove that the set $\{x_n\}$...
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### Importance of a basic sequence in Banach Space Theory

In Classical Banach Spaces I and II by Lindenstrauss and Tzafriri, their first definition in page $1$ is as follows: A sequence $\{x_n\}_{n=1}^\infty$ in a Banach space $X$ is called a Schauder ...
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### What generates $\ell^\infty$?

I am reading a accepted answer here. But I don't know why $\phi(C[0,1])$ generats $\ell^\infty({\mathbb{N}} )$ or what generates $\ell^\infty(\mathbb{N})$ as a von Neumann algebra. Can someone tell ...
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### Schauder basis in a non-normable space

I want to better understand what characterize the vector spaces that have a Schauder basis. Usually such a basis $\{u_i \} \quad i\in \mathbb{N}$ is defined for a Banach space using the norm to ...
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### Show that $(X,|||\cdot|||)$ is a Banach space.

In the book 'Classical Banach Spaces I and II' by Lindenstrauss, page $1,$ Chapter $1$, he stated the following: Let $(X,\|\cdot\|)$ be a Banach space with a (Schauder) basis $(x_n)_{n=1}^{\infty}.$...
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### Singular value decomposition of a specific integraloperator

I want to determin the singular value decomposition of the integral operator $$L^2(0,1) \to L^2(0,1) ; f \mapsto Af(\cdot) = \int_0^\cdot f(y)dy.$$ Its adjungate is given by A^*f(\cdot) = \int_\...
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### Does c have a schauder basis?

Let $c=\{(a_n) : a_n \in \mathbb{R}, \lim_n a_n = L \operatorname{exists} \}$ be equipped with the sup-norm. Does $c$ have a Schauder basis? $c$ is a separable space . That is why my first ...
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### Schauder basis are dense in any Hilbert space

Definition: A sequence $(x_n)$ in a Hilbert space H is called total iff for any $x\in H, \quad$ $<x,x_n>=0$ implies $x=0$. I want to show that any schauder basis $(e_n)_{n=1}^{\infty}$ in a ...
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### Mazur's Weak Basis Theorem

It is the Exercise 1.1 in Topics in Banach Space Theory by Albiac and Kalton to prove Mazur's Weak Basis Theorem, which states that every weak basis in a Banach space $X$ is a Schauder basis, where ...
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### All $e_i's$ are in $l_1$ but $e_1+e_2+…+e_n+…$ is not in $l_1$, why? Hint for the the pattern of basis for $l_1$ space.

Let us consider $l_1$ space with the norm $||x||=\sum_{i=1}^\infty|x_i|$ where $x=(x_1,x_2,...,x_n,...)$. Now let us take $e_1=(1,0,0,...$), $e_2=(0,1,0,0,...$),$...$,$e_n=(0,0,0,...,1,0,0,...)$. ...
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### Existance of Schauder Basis of vector space over field F implies separability

It's rather trivial to prove that when a normed vector space X over the real numbers, or the complex numbers, has a Schauder Basis then X is separable. Since you can construct a dense&countable ...