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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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basis in Hilbert spaces

Let $H$ be a separable Hilbert space and $(x_n)$ a sequence with dense span. This is the standing assumption for the whole discussion. Now consider the two possible additional assumptions: (a) $\{ x_n:...
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Linear subspace of Infinite linear combination, closed or not?

Let $\{Y_{n}\}_{n=1}^{\infty}$ be a sequence in an infinite-dimensional Lebesgue space $L^{p}(\mathsf{\Omega},\mathcal{F},\mathrm{P})$, $1<p<\infty$, on some probability space $(\mathsf{\Omega},\...
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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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Limit operations and Schauder combinations

Let $V$ be a Banach (Hilbert if needed) space with a countable (orthonormal if needed) Schauder basis $\{v^i:i\in\mathbb{R}\}$. Does this basis always work like the Fourier basis $\{1,\cos(x),\sin(x),\...
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Must the vector vi coordinates with respect to the Schauder basis $e_1, e_2, e_3, ..., e_n$,necessarily approach zero as i increases to infinity?

If $e_1, e_2, e_3, ..., e_n$,are the Schauder basis of the normed space (E, ||.||) with the property that $||en|| = 1$ for all $n ∈ N$, and $(v_i)i∈N ⊂ E$ is a sequence such that $lim_{i→∞} ||v_i|| =...
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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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Compactly supported orthonormal basis of $L^2(\mathbf R)$ with certain properties

Take a positive integer $\alpha$. I am looking for an orthonormal basis $(\phi_n)$ for $L^2(\mathbf R)$ with roughly speaking the following properties: each $\phi_n$ is compactly supported and $C^\...
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Notion of a basis that allows infinite sums

So with no other structure it is clear that a vector space has no notion of convergence and thus infinite sums make no sense. However, suppose $V$ is an inner product space, then we have a norm which ...
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Find subset of vectors which form basis

Question Let W be the subspace of $R^5$ spanned by$ u_1 = (1, 2, –1, 3, 4)\\ u_2 = (2, 4, –2, 6, 8) \\ u_3 = (1, 3, 2, 2, 6)\\ u_4 = (1, 4, 5, 1, 8)\\u_5 = (2, 7, 3, 3, 9)$ Find a subset of the ...
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Set of uniqueness of a harmonic series

Consider the family of functions $$ \forall n \in \mathbb N,\quad \forall x \in \mathbb R,\quad e_n(x) = e^{in x}, $$ which is known to be a Riesz basis of $L^2(0,2 \pi)$. What happens when we look at ...
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Is a sequence in a Hilbert space that forms biorthogonal with a Schauder basis also a Schauder basis?

Suppose $H$ is a Hilbert space and has a Schauder basis $\{e_{j}\}_{j=1}^{\infty} $. $\{\hat{e_{j}}\}_{j=1}^{\infty}$ is a sequence biorthogonal with $\{e_{j}\}_{j=1}^{\infty}$; that is, $<e_{j},\...
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Showing that $\{\varphi_{m,n} \}_{m \geq 1, n \geq 1}$ is an orthonormal basis for $L^2((a,b) \times (a,b)).$

Let $(a,b) \subseteq \mathbb R$ and $\{\varphi_n \}_{n \geq 1}$ be an orthonormal basis for $L^2((a,b)).$ Define $\varphi_{m,n} : (a,b) \times (a,b) \longrightarrow \mathbb C$ by $$\varphi_{m,n} (s,t) ...
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Are spreading sequences uniformly spreading?

Let us say that two sequences $(x_n)_n$ and $(y_n)_n$ in (possibly different) Banach spaces $X$ and $Y$ are $\lambda$-equivalent, with $\lambda\geq 1$, if for every sequence of coefficients $(a_n)_n\...
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Representation of semigroup of trace-class operators in terms of projectors

Let $H$ be a Hilbert space. For all $t\in\left]0,\infty\right[$ the operator $A_t\in L(H,H)$ is assumed to be trace-class and symmetric. Furthermore $A$ is a semigroup, i.e. $$A_{t+h}=A_tA_h$$ for all ...
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Can an eigenbasis be divided into bases of the eigenspaces?

I am considering an operator $A$ defined on a subspace of some Hilbert space $H$ and I am given a Hilbert-basis $(v_n)_{n\in\mathbb N}$ such that all $v_n$ are eigenvectors of $A$: $$Av_n=\lambda v_n$$...
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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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On the "basis" of the space $C^\infty(a,b)$ of smooth functions

My math methods textbook (Hassani's Mathematical Physics) makes the following (I think dubious) claim: If we assume that a < 0 < b, then the set of monomials $1,x,x^2,...$ forms a basis for $C^∞...
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Special orthonormal basis for space of continuous real functions on a closed interval

Let $a,b \in \mathbb{R}$ with $a < b$. Let also $P = \{x_1, \ldots, x_n\}$ (with $n > 1$) be a finite subset of $[a,b]$ with all distinct elements ($x_1 < x_2 < \cdots < x_n$). Let $C[a,...
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A normed space with a Schauder basis is separable.

I am trying to prove the theorem: A normed space $X$ with a Schauder basis is separable. I tried to do it the following way. Let $X$ be a real space. Suppose that $(e_n), e_n\in X\,\forall n\in \...
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Subset of a basis for a normed vector space cannot be Cauchy?

In class, we showed that $e_n \in l^p$ converges weakly but not strongly for $p \in (1,\infty)$ and that $\sin(2\pi nx) \in L^2([0,1])$ converges weakly but not strongly. I was wondering if basis ...
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Are Schauder-like uncountable bases possible and of unique cardinality?

Let me clarify what I mean by "uncountable" here, which differs from some other Q's with similar titles here. I mean the analogue of a Hamel basis, but allowing countable sums instead of ...
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Transfinite Ordinal Schauder Bases

It is known that not every separable Banach space has a Schauder basis, and certainly, non-separable Banach spaces cannot. This got me thinking about generalising Schauder bases to work for non-...
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On showing that the usual basis $(e_n)_{n \in \Bbb N}$ is not Schauder for $\ell^\infty.$

I am aware that there is more than one post about this in the forum already. Altought, I did a proof by myself and I have some doubts midway! Too keep in mind: here, I am trying to prove that the ...
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Why do we prefer the Schauder basis over the Hamel basis in functional analysis?

Our functional analysis instructor mentioned in the class that the Hamel basis is not so important in the context of Banach spaces. Instead, we prefer the Schauder basis. However, he did not specify ...
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Is there an exposition of the space of Per Enflo?

Famously, a problem which stood for many years concerning Banach spaces was whether or not every one admitted a Schauder basis. This was discussed by many mathematicians, including Grothendieck! ...
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Proving standard basis of $l^p$, $1<p<\infty$, to be shrinking

The definition of shrinking basis is given inside the Wikipedia article, under the section "Schauder basis and Duality". It says that "if $1<p<\infty$, then the standard basis $(...
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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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Zero diagonal operators and Riesz bases

We shall concern ourselves only with separable complex Hilbert spaces and bounded linear operators on them. An operator $A: \mathcal H\to \mathcal H$ is said to be a zero diagonal operator if $(Ae_n, ...
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Can we always construct a Schauder basis incorporating finite numbers of given linearly independent vectors?

Assume $\mathcal{H}$ is a separable Hilbert space with orthonormal basis function $\{e_i\}_{i=1}^{\infty}$ that are countably infinite. We know there always exists a Schauder basis $\{ y_i \}_{i=1}^{\...
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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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Canonical basis in $l^{\infty}$

For an exercise, I need to show that the canonical basis is not a valid basis in $l^{\infty}$. Concretely, the exercise states : Consider the Banach space $l^{\infty}$ of sequences $x = \{x_n\}_{n = 1}...
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Schauder basis and convex combinations

Let us suppose that $(X, \lVert \cdot \rVert)$ is a normed space over $\mathbb{R}$ which has a Schauder basis, that is, there is a sequence of vectors $(x_n)_n$ in $X$ such that for all $x \in X$ ...
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Is a Schauder basis countable [duplicate]

Does the definition of Schauder basis of a Banach space include that it be countable. My issue is a theorem that states a Hilbert space is separable if and only if it has a countable orthonormal basis....
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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}...
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prove that a space with a Schauder basis is separable, that is, it contains a countable dense subset.

Q: I can't get the sketch of the proof Now since an arbitrary element of $X$ having the expansion $\sum_{n=1}^\infty \alpha_n b_n$ in terms of $(b_n)_{n \in N}$ can be written as $\lim _{N \rightarrow ...
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Orthonormal Basis of Bergman Space

This is a problem (#39 or #40 depending on the edition) at the end of Chapter 1 in Krantz's book Function Theory of Several Complex Variables. Let $\Omega\subset\mathbb{C}^n$ be a smooth and bounded ...
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Give a example to show that there not necessarily exist uniformly bounded linear funtionals $\{f_j\}$ such that $f_j(x_k)=\delta_{jk}$.

Suppose $E$ is a normed linear space, $\{x_k\}_{k=1}^{\infty}$ is linearly independent and $\|x_k\|=1, k=1,2,\cdots$. Give a example to show that there not necessarily exist uniformly bounded linear ...
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How does one show a Schauder basis is shrinking?

I feel like I must be missing a trick - I'm self studying functional analysis and have come across Schauder bases, and I'm looking at different classifications e.g. shrinking, boundedly complete, ...
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Schauder decomposition of a Banach space E

From book J. T. Marti, Introduction to the Theory of Bases, 2013 and I. Singer, Bases in Banach Spaces II, 2011. Theorem For a Banach space $E$ the following statements are equivalent: There is a ...
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Adjoint system associated to Schauder basis

Let $X$ be reflexive Banach space or if you need one could assume that $X=H$ where $H$ is a usual separable Hilbert space and $\{e_n\}_{n = 1}^\infty$ be a Schauder basis in it which means for any x $\...
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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 ...
Roba's user avatar
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The set of vectors {a, b, c} is a basis of R³. Determine if another set, {3a-2b, a+4b+5c, a-2c}, is also a basis of R³.

Since the set of vectors $\{\vec{a},\vec{b},\vec{c}\}$ are a basis of $\mathbb{R^3}$, we can understand that they are linearly independent. This means that: $\alpha_1\cdotp\vec{a}+\alpha_2\cdotp\vec{...
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Basic subsequence of weak* convergent sequence

I have given the following theorem : Let $(\phi_n)$ is a sequence on $S_{X^*}$ (unit sphere of topological dual of the Banach space $X$). If $(f_n)$ is weak* convergent to zero then $f_n$ has a basic ...
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