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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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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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Every basis in a finite dimensional Hilbert space is a Riesz Basis

The goal is to prove that every basis in a finite-dimensional Hilbert space is a Riesz basis, i.e., there exist constants $A>0$ and $B>0$ for the basis $\{x_k\}$ such that: $$ A \sum_n |a[n]|^2 \...
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Uncountable basis in Hilbert space vs orthonormal basis

It is known that an infinite dimensional Banach space does not have a countable Hamel basis. It is also known that a separable Hilbert space has an orthonormal countable basis. Now, I think this basis ...
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Is a von Neumann algebra a closed linear span of pairwise orthogonal projections?

It's well known that a von Neumann algebra is a closed linear span of its projections. Can we require these projections to be pairwise orthogonal? that is, can we find a set $\mathscr P$ ...
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Intersection of spaces with Schauder basis

Let $\{v_n\}_{n \in \mathbb{N}}$ be a basic sequence in $\ell^2$ over $\mathbb{C}$ Let $V_m =\overline{\operatorname{span}} \{v_n\}_{n \geq m} $ Let $\{u_n\}_{n \in \mathbb{N}}$ be a basic sequence ...
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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 $...
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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 ...
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Does $C_0$ have a Schauder Basis.?

I have proved (i) by showing that $C_0$ is closed in $\ell_{\infty},|| -||_{\infty}$ (ii) was proven by showing that it is sequentially compact and using some diagonalization argument. But I'm stuck ...
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How can I get transformed coordinate between different basis?

For example, there is set of DCT basis which are orthogonal: $F_1(x,y),F_2(x,y)\cdots,F_N(x,y)$. So, given function can be uniquely expressed as sum of DCT basis. (we decompose an image to sum of DCT ...
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(Weak) basis for the space of bounded sequences

The canonical basis is not a Schauder basis of the space of bounded sequences, but in some way, it uniquely determines every element in the space. Is it a basis in a weaker sense? How is it called? ...
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normed space which posses a countable algebraic basis that a Banach space cannot posses such a basis

I have have read some papers related to normed space , it came to mind to Find a normed space wchich posses a countable algebraic basis that a Banach space cannot posses such a basis , But i failed to ...
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Unconditional constant vs Suppression unconditional constant

I am trying to solve the following question. Let $(u_n)_{n=1}^{\infty}$ be an unconditional basis for a Banach space $X$ with suppression-unconditional constant $K_{su}$. Prove that for all $N$, ...
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Schauder basis under isometry.

I am self studying linear functional analysis and am a bit confused about the following problem. It is situated after a chapter on the open mapping and closed graph theorem and in my answer I don't ...
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Can the existence of a Schauder basis help in the verification of the convergence of a sequence of operators?

I was wondering if the existence of a Schauder basis helps in the verification of the convergence of a sequence of operators. The following is straightforward: Let $(T_n), T\in \mathcal{L}(X)$, ...
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42 views

Coefficients of Schauder basis do not change on rearrangement

Let $X$ be a Banach space (maybe even a normed space is enough). Suppose that $(\psi_i)_{i=1}^\infty \subseteq X$ is a Schauder basis and let $\sigma:\mathbb{N} \to \mathbb{N}$ be a bijection such ...
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38 views

Schauder basis in $C([0,1],H)$ where H is a Hilbert space

Take a real separable Hilbert space $H$. Consider the space $C([0,1],H)$, i.e. the space of continuous functions $[0,1] \to H$, endowed with the sup norm. Does $C([0,1],H)$ admit a Schauder basis?
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Basic for a specific function space

I know that in general does not exist a countable basis for functions in the form $f : \mathbb{N} \rightarrow \mathbb{N}$. But if I restrict to the functions $f(k) = (k)_n$, for fixed $n = 0, 1, 2, \...
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Is the sequence $\{\frac{1}{n}e_n\}_{n \in \mathbb{N}}$ a Schauder basis of $\ell^2$ over $\mathbb{C}$

Let $\{e_n\}_{n \in \mathbb{N}}$ be the standard basis of $\ell^2$ over $\mathbb{C}$ Let $\{b_n\}_{n \in \mathbb{N}}$ be the sequence such that $\forall n \in \mathbb{N}: b_n = \frac{1}{n}e_n$ My ...
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$f_k := e_k + e_{k+1}$ complete, but $e_1 \neq \sum_{k=1}^{\infty} \alpha_k f_k$ for any $\alpha_k$

Let $\mathcal H$ be a Hilbert space and $\{e_k\}_{k=1}^{\infty}$ a orthonormal basis for $\mathcal H$. Consider $f_k := e_k + e_{k+1}$, $k \in \mathbb N$. Claim: $\{f_k\}_{k=1}^{\infty}$ is complete, ...
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Is the Unit ball of $X^{**} $ weak*- sequentially compact?

Am in the middle of a problem and i have the following conditions : Let $X$ be a reflexive Banach space with Schauder basis $(e_n)_{n=1}^{\infty}$, i have a sequence $x_n^{**} \in B_{X^{**}}\biggl(...
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Interchangability of arbitrary sums and linear operator

Let $\mathcal H$ be a Hilbert space, $\{x_i : i \in I \}$ be a orthonormal base in $\mathcal H$ and $T \in L(\mathcal H)$. Does the following hold: $T( \sum_{i \in I} \lambda_i x_i) = \sum_{i \in I}...
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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 $v\in V$ ...
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Is a Reproducing Kernel Hilbert Space just a Hilbert space equipped with an “indexed basis”?

I haven't studied any functional analysis yet. My linear algebra is pretty good, I think. Consider the tuple $(H, I, \phi)$ where $H$ is a Hilbert space, $I$ is an abstract set, and $\phi:I \to H$ is ...
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Solving equation with orthonormal basis in it

Let $(v_n)_n$ be an orthonormal base of $\mathcal H$ Hilbert space, $(\sigma_n)_n \subset \mathbb R$ and $x,y \in \mathcal H$. Is $\sum_{n=1}^{\infty} \langle x, v_n \rangle \sigma_n^2v_n = y$ if ...
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170 views

An orthogonal basis of a Hilbert space is Schauder?

I read that an orthogonal (Hamel) basis $(e_i)$ of a Hilbert space is always a Schauder basis. I can see why; if $\sum \alpha_i e_i=0$ then taking the inner product with each $e_i$ gives $\alpha_i =0$....
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136 views

Does every (non-separable) Hilbert space have the approximation property?

We can prove that every Banach space with a Schauder basis has the approximation property. I've read that this implies that every Hilbert space $H$ has the approximation property. It's clear to me ...
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Schauder basis $L^p(\mathbb{R})$

Let $\{e_{n}(x)\}_{n=0}^{\infty}$ be orthonormal basis of Hilbert space $L^2(\mathbb{R})$. If $\{e_{n}(x)\}_{n=0}^{\infty} \subset L^p(\mathbb{R})$ for some $p\geq 1$, is the $\{e_{n}(x)\}_{n=0}^{\...
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119 views

Finding a Schauder basis in a countable generating subset without Zorn's lemma

Let $V$ be a (Hausdorff) topological vector space, and let $S = \{v_i\} _{i \in \mathbb N} \subset V$ be a generating subset (notice that it is countable!). Does countability make it possible to ...
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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 ...
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1answer
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Unconditional basis and the Banach-Steinhaus theorem

A sequence $\{x_k\}$ in a Banach space $X$ is a Schauder basis of $X$ if every element $x\in X$ has a unique representation $$ x = \sum_{k=1}^\infty c_k x_k $$ with the series converging in the norm ...
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Existence of a Basis in a Banach Space

Suppose $X$ is a Banach space, $\left \{ e_n \right \}$ a sequence in $X,\ \overline {\text{span}\left \{ e_n \right \}}=X$ such that $e_n\neq 0$ and such that there is a constant $K$ with the ...
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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\|...
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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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$(e_1,e_2,..)$ is not a Schauder basis of $\ell^\infty$ [duplicate]

Show that $(e_1,e_2,...)$ is not a Schauder basis of $\ell^\infty$ where $e_i$ is the vector in $\mathbb R^\infty$ with 1 in the ith coordinate and 0 elsewhere and $\ell^\infty=\{(x_1,x_2,...)|x_i\in \...
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103 views

Unconditional Schauder Basis Convergence?

Let $X$ be a Banach space and $\{e_n\}$ be a unconditional Schauder basis for $X$, i.e., if for every $x \in X$, its expansion $$x = \sum_i a_i e_i$$ converges unconditionally. (Recall that a ...
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59 views

Is $\lim_{i \to \infty}e_i$ in $ c_0$?

$lim_{i \to \infty}e_i$ is a sequence which is not convergent, where $e_i$ denote the sequence whose $i$th term is 1 and rest all are zero. From this point of view, we can say $\lim_{i \to \infty}e_i\...
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$x=\lim_{i \to \infty}e_i$ is a sequence which is not convergent, so $x\notin c_0$, then why {$e_i$}$_{i=1}^\infty$ is a Schauder basis for $c_0$?

Let $e_i$ denote the sequence whose $i$th term is 1 and rest all are zero. Then $x= lim_{i \to \infty}e_i$ is a sequence which is not convergent. So $x\notin c_0$ and so $x\notin l_p$ because $l_p\...
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1answer
42 views

Unbounded adjoint

I am learning about adjoints of unbounded operators on infinite-dimensional Hilbert spaces. I understand that the domain of such an adjoint (to operator $A$, from Hilbert space $H$ onto itself) is ...
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1answer
51 views

Absolute values of coordinates (basis representation).

It looks very simple, but I couldn't check this. Let $(X, \|\cdot\|)$ be a vector space of dimension $n$ (over $\mathbb{R}$ or $\mathbb{C}$), where $\| \cdot \|$ is a norm on $X$. Let $\{e_1, \...
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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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78 views

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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76 views

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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1answer
265 views

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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215 views

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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135 views

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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37 views

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 ...