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Questions tagged [schauder-basis]

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

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What is the difference between a Hamel basis and a Schauder basis?

Let $V$ be a vector space with infinite dimensions. A Hamel basis for $V$ is an ordered set of linearly independent vectors $\{ v_i \ | \ i \in I\}$ such that any $v \in V$ can be expressed as a finite linear combination of the $v_i$'s; so $\{ v_i \…
Lor
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How to prove that if a normed space has Schauder basis, then it is separable? What about the converse?

Can we take as a dense subset the collection of all the linear combinations of the vectors of the Schauder basis using the rationals as scalars (or the complex numbers with rational real and imaginary parts for that matter)? What can we say about…
Saaqib Mahmood
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Is there a notion of a continuous basis of a Banach space?

If $X$ is a Banach space, then a Hamel basis of $X$ is a subset $B$ of $X$ such that every element of $X$ can be written uniquely as a linear combination of elements of $B$. And a Schauder basis of $X$ is a subset $B$ of $X$ such that every element…
Keshav Srinivasan
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Why isn't every Hamel basis a Schauder basis?

I seem to have tripped on the common Hamel/Schauder confusion. If $X$ is any vector space (not necessarily finite dimension) and $B$ is a linearly independent subset that spans $X$, then $B$ is a Hamel basis for $X$. If there exists a sequence…
Fequish
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Are uncountable "Schauder-like" bases studied/used?

We could define the following notion of basis in a way analogous to unconditional Schauder basis: If $X$ is a topological vector space over $\mathbb R$ and $B=\{b_i; i\in I\}$ be a subset of $X$. We say that $B$ is a basis if, for every $x\in X$…
Martin Sleziak
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Do reflexive separable Banach spaces have Schauder bases?

I know there exists an example in the literature due to Per Enflo of a separable Banach space without a Schauder basis. I am wondering if there is a reflexive counterexample? If so I would greatly appreciate a reference.
Ben C
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Does the ordering of a Schauder basis matter in Hilbert space?

If $S=\{v_i\}_{i\in\mathbb N}$ is a (not necessarily orthogonal) Schauder basis for a Hilbert space $H$, must $S$ be an unconditional Schauder basis? I define these terms below because not every source I have found agrees perfectly on the…
WillG
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Normalizing a semi-normalized Schauder basis

Here's something that is probably obvious but I can't seem to see it. Suppose $(x_n)_{n=1}^\infty$ is a Schauder basis for a Banach space $X$, "seminormalized" in the sense that we have $$0<\inf\|x_n\|\leq\sup\|x_n\|<\infty.$$ Now consider the…
Ben W
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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 weak basis is defined as follows: A sequence…
Vitor Borges
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Schauder basis that is not Hilbert basis

Given an infinite dimensional Banach space $(V,\|\cdot\|)$ over the field $\Bbb K=\Bbb C$ or $\Bbb R$, a countable ordered set $B:=\{b_n\}_{n\in\Bbb N}⊂V$ is called Schauder basis, if every $v\in V$ can be uniquely decomposed as: $$…
giobrach
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What do we call a Schauder-like basis that is uncountable?

In a topological vector space, every Schauder basis is assumed countable, by definition. Supposing we drop the countability condition, we call this a [what goes here?] basis?
goblin GONE
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Definition of Schauder basis

I have a definition of a Schauder basis but I’m unsure of it. The definition I have is A sequence $\{e_k : k \in \mathbb{N} \}$ in a normed space $(V, \| \cdot \| )$ is a Schauder basis if $\sum_{k=1}^{\infty } \alpha _k e_k =0 $ implies $\alpha _k…
Anonmath101
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Is a linearly independent set whose span is dense a Schauder basis?

If $X$ is a Banach space, then a Schauder basis of $X$ is a subset $B$ of $X$ such that every element of $X$ can be written uniquely as an infinite linear combination of elements of $B$. My question is, if $A$ is a linearly independent subset of…
Keshav Srinivasan
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Which topological vector spaces have uncountable unordered sums?

If $P$ is an uncountable locally finite poset, then the incidence algebra $I(P)$ is a topological vector space (in fact a topological algebra) with the interesting property that every element $f$ can be written uniquely as an uncountable unordered…
Keshav Srinivasan
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Constructively embedding $\mathbb{Q}^\mathbb{N}$ into $\mathbb{R}$

Using the axiom of choice it is provable that $\mathbb{R}$ is isomorphic to $\mathbb{Q}^\mathbb{N}$ as a vector space over $\mathbb{Q}$. (Assuming AC, both spaces have a Hamel basis over $\mathbb{Q}$ of the same cardinality and are thus…
H.D. Kirchmann
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