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Questions tagged [schauder-basis]

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

229 questions
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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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4 answers

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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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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2 answers

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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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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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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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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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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Can the trace be computed in any Schauder basis?

Edit: This question didn't get any answers after a bounty, so I cross-posted it here on Math Overflow. Let $H$ be a separable Hilbert space and $T \in L(H)$ a trace-class operator. It is well known that the trace of $T$ is given by…
WillG
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2 answers

Can all linear operators be represented as matrices? (In infinite dimension.)

Old title: Components of unbounded endomorphism wrt Schauder basis? Changed for discoverarbility. In finite-dimensional linear algebra, given an endomorphism $T$ and a basis $\boldsymbol{e}_i$, we can easily find a matrix representation of $T$: $$ …
csha
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Basis of $C[a,b]$

The space $C[a,b]$ , space of all real valued continuous functions on $[a,b]$ is an infinite dimensional vector space over the field $\Bbb R$. As every vector space over a field has a basis so definitely $C[a,b]$ has a basis. I want to know a basis…
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