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Questions tagged [hamel-basis]

A Hamel basis (often simply called basis) of a vector space $V$ over a field $F$ is a linearly independent subset of $V$ that spans it.

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Is a linear map determined by the image of an orthonormal basis?

I have just recently started learning about infinite dimension vector spaces: in particular, Hilbert spaces. I have read about the concept of an orthonormal basis, the difference with respect to a ...
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Basis of $\mathbb{R}$ over $\mathbb{Q}$ exists by Axiom of choice, but is it impossible to construct it and its cardinality?

Is it hard or proven to be impossible to construct basis $B$ of $\mathbb{R}$ over $\mathbb{Q}$? Small question regarding the cardinality: (If some miracle happened and CH turned out to be false, then ...
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X is a vector space such that dim(X)=n. $M \subset X $ is a subspace. let $B=\{b_1,b_2,...,b_m\} $ is a hamel basis for M.

if X is a vector space such that dim(X)=n and $M \subset X $ is a subspace. let $B=\{b_1,b_2,...,b_m\} $ is a hamel basis for M. then $dim(M)\le dim(X) $ there is a set $ D=\{d_1,d_2,...,d_{m-n} \}$ ...
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Confusion regarding orthonormal basis of $L^2[0, 1]$, in requiring $f(0) = f(1)$?

Let us consider here the continuous elements of $L^2$. It is often stated that the family $e_k(x) = e^{-2 \pi i k x}$ is an orthonormal basis of $L^2[0, 1]$, in that a function can be written as $$ f(...
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Brezis 5.28.2 About the competition of an orthogonal sequence

Let $(e_n)$ be an orthonormal sequence in $H$ is parable. Prove that there exists an orthonormal basis of $H$ that contains $\bigcup_{n\geq 1} \{e_n\}$ My try: Define the set $\Omega$ as follows: $$\...
Est's user avatar
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Basis for an infinite dimensional vector space

After learning about finite dimensional vector spaces, the time came for learning about infinite dimensional ones. However, these seem much less intuitive to me, and proof of that is the question I ...
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Does "every normed vector space has a basis" imply choice

It is known that if every vector space has a basis, then the axiom of choice holds. Is the weaker claim that every normed space (over $\mathbb{R}$ or $\mathbb{C}$) has a basis enough to prove $AC$? I'...
Ynir Paz's user avatar
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Definition of an infinite dimensional vector space without the axiom of choice

A finite dimensional vector space is a vector space which has a basis (linearly independent spanning subset) with finite cardinality. If we accept the axiom of choice then it can be proven that every ...
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If a basis for $V$ intersects an $m$-dimensional subspace of $V$ at $m$ basis vectors, will these $m$ vectors form a basis for the subspace?

Sorry, I might be asking a stupid question, but somehow I can't convince myself of the assertion in the title: Let $\{u_1,\ldots,u_n\}$ be a basis for a vector space $(V,F)$ and suppose $S$ is an $m$-...
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Finite Dimensional Subspace of a Normed Linear Space is complete.

I am working through the wiki proof that every finite dimensional subspace of a normed linear space is closed. The proof goes as follows: Let $(V ,\| \cdot \|)$ be a normed linear space. Let $W$ be a ...
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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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Does the origin have a strictly convex bounded neighborhood?

Given any finite-dimensional normed space, the topology is equivalent to that generated by the Euclidean norm. So the Euclidean open ball is norm closed. The fact that is is strictly convex is ...
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Reproducing kernel Hilbert space induced by $k(x, y) = \delta_{x, y}$, where $\delta$ is the Kronecker delta

I am trying to find the reproducing kernel Hilbert space induced by the symmetric positive definite (and bounded and measurable) kernel $$ k \colon X \times X \to \{ 0, 1 \}, \qquad (x, y) \mapsto %\...
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Is there an infinite dimensional inner product space without an orthogonal Hamel basis?

I want to know if there exists an (real or complex) vector space $X$ with infinite dimension and an inner product $\langle\cdot,\cdot\rangle$ such that there is no orthogonal Hamel (algebraic) basis ...
Victor Ronchim's user avatar
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Is the alternative proof correct? if not why?

(If $\ker f\subset \ker g$ where $f,g $ are non-zero linear functionals then show that $f=cg$ for some $c\in F$.) I came across such a task in the textbook only with the condition if $kerf= kerg$ $V = ...
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Can any spanning subset be reduced to a Hamel basis in infinite dimensions?

As per this question, it is possible to extend any l.i. subset of vectors $A \subset V$ of a vector space to a (Hamel) basis, $B \subset V$. Is it likewise possible to delete vectors from a spanning ...
EE18's user avatar
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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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Immediate consequence of a theorem related to spanning set/basis - linear algebra

I am going through a theorem and its corollary from a text book: Theorem $T_0$: Let $W$ be a subspace of $R^n$, and let $B = \{v_1,...,v_p\}$ be a spanning set of $W$ containing $p$ vectors. Then any ...
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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,...
Alberto Carraro's user avatar
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Fundamental polynomials: do they form a base

First, is $P_n$ $n$ or rather $n+1$ dimensional real vector space of polynomials of degree at most $n$ ? Here, $n$ indicates that it should be $n$ but the basis $1,x,x^2,x^3,...,x^n$ has $n+1$ ...
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A proof that the null linear mapping is the only one whose matrix representation does not depend on the basis [FALSE]

I would like to show the fact that the linear mapping $$ L : E\to E $$ $$ x\mapsto0_{E} $$ is the unique linear mapping whose matrix representation does not depend on the choice of the basis. My ...
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Basis criterion for vector space

I would like to show the following theorem : Let $E$ be a vector space and $S=\{s_i : i\in I\}\subset E$. Then we have the equivalence $S$ is a basis of $E$ For any function $f : S\to E$ there exists ...
G2MWF's user avatar
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Prove that the vector space of real convergent power series does not have a countable basis

Define $V$ as the subset of $\mathbb{R}[[x]]$, the $\mathbb{R}$-vector space of formal power series with real coefficients, such that for any $f\in V$, for any $r\in\mathbb{R}$, the series $f$ always ...
durianice's user avatar
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Find basis of intersection of 2 spans with unequal dimensions

I've been stuck on this question for quite a while: Given $U =$ span $\left\{ \begin{pmatrix} 0\\ 2\\ 0\\ 0 \end{pmatrix}, \begin{pmatrix} 1\\ 0\\ 0\\ 0 \end{pmatrix}, \begin{pmatrix} 2\\ 1\\ 3\\ 7 \...
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Can a infinite dimensional Banach space have a dense Hamel basis [duplicate]

I'm looking for a link of a book or paper that has the following Theorem: Let $X$ be an infinite dimensional Banach space. Then there exists a Hamel basis $B$ of $X$ that is dense in $X$ ( That is $\...
lebong66's user avatar
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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 ...
Kishalay Sarkar's user avatar
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Hamel basis of $\ell_p$ and $\ell_\infty$.

Consider the space $\ell_p=\{x=\{x_n\}: \sum|x_n|^p<\infty\}$ for $1\leq p<\infty$.We know that $\ell_p$ can be given $\|.\|_q$ for $q>p$ which is not equivalent to $\|.\|_p$,the usual norm....
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Proof that a linear map from infinite dimensional vector space to finite dimensional vector space can't be injective

In other words, let $T: V \to W$ be an injective linear map. Show that if $V$ is infinite-dimensional then $W$ must also be infinite-dimensional. I think the way I want to approach this is to show ...
Oliver's user avatar
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Uniqueness of basis of a vector space

If X is a vector space and: $A \subset X$ s.t. $span(A)=X $ ( A basis of X ). If $x \in span(A)$ : $x= \sum_{i=1}^{n} x_i k_i$ for $x_1,...,x_n \in X, k_1,...,k_n \in \mathbb{R}$. How to prove that $...
lebong66's user avatar
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Algorithms for extracting a basis

Let us consider a rectangle matrix $A$ in $M_{mn}(\mathbb{C})$ with $m<n$. Suppose that $\operatorname{rank}A=m$. What are the well-know algorithms to extract $m$ linearly independent columns of $A$...
ABB's user avatar
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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 ...
Paul's user avatar
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Hamel basis and real algebraic numbers

Is the following picture correct Mainly: is it true that a Hamel basis of $\mathbb{R}$ is a subset of $\mathbb{R}$ and that it contains the real algebraic numbers? If so, is there any reference for ...
sam wolfe's user avatar
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Understanding the proof of "If $V$ is $n$ dimensional vectors space then a set in $V$ with fewer than $n$ vectors does not span $V$ "

In below image author has proved the theorem which states that, "Let $V$ ve a finite dimensional vector space and $\{v_1,v_2,...,v_n\}$ is basis for $V$ then, "If a set in $V$ has fewer than ...
Akash Patalwanshi's user avatar
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1 answer
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Relation of Hamel basis with the equation $f(x + y) = f(x) + f(y)$? [duplicate]

I am reading "Linear and Nonlinear Functional Analysis with Applications by Philippe G. Ciarlet ", which explains the origin of hamel basis by a problem: Describe the set $F$ of all ...
Chandler's user avatar
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Where I went wrong in this chain of arguments - Linear algebra

Let $AB = I_n$. $A$ and $B$ are nonsingular, square matrices of size $n$. Let $A_{r1}$ be the first row of $A$. The products $A_{r1} B_j = 0, j \in \{2,\dots,N\}$. $B_j$ is the $j^{th}$ column of $B$. ...
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Total orthonormal set which is not a basis [duplicate]

Does there exist a total orthonormal set in a Hilbert space which is not a basis? In a separable Hilbert space every total orthonormal set is a basis. What if the Hilbert space is not separable?
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What is an example of spanning set with linearly independent vectors that is not a Hamel Basis?

While trying to distinguish the definitions of Hamel and Schauder bases of infinite dimensional vector spaces, I was told that the true definition of a Hamel basis is not "a spanning set with ...
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Prove that the algebraic dimension of an infinite dimensional Banach space is atleast $\mathfrak{c}$.

Question: Prove that the algebraic dimension of an infinite dimensional Banach space is atleast $\mathfrak{c}$. Proof: $(X, \|•\|) $ be a infinite dimensional Banach space. Then $\dim(X) \ge \aleph_{0}...
Ussesjskskns's user avatar
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2 answers
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Given a basis $X$ for a finite dimentional vector space $V$, prove that $L(X)$ spans $L(V)$.

Let $L : V \to W$ be a linear map, and suppose $X := \{v_{1},\ldots, v_{n}\}$ is a basis for $V$. Prove that $Y := \{L(v_{1}),\ldots, L(v_{n})\}$ is a spanning set for $L(V)$. So I am a struggling on ...
Joe's user avatar
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Give a example to show that there not necessarily exist $\{e_j\}$ such that $f_i(e_j)=\delta_{ij}, i,j\geq 1$.

Suppose $X$ is a normed linear space. $\{f_i\}\subset X^*$ are linearly independent. Give a example to show that there not necessarily exist $\{e_j\}$ such that $f_i(e_j)=\delta_{ij}, i,j\geq 1$. I ...
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If $S$ is a spanning set of a vector space, must there be a subset of $S$ which is a basis?

Let $V$ be a finite-dimensional vector space. Is it possible that a set $S$ spans $V$, but no subset of $S$ is a basis of $V$? I think that the answer is no, at least in the case that $S$ is finite. ...
Joe's user avatar
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Difficulty understanding the link between the span of a sequence being an isomorph of $\ell_1$ and that sequence being a basis of the **closed** span

$\newcommand{\span}{\operatorname{span}}$EDIT: By some further reading, I note that the very fact that $\{x_n\}$ does not contain a weak convergent subsequence implies by Rosenthal's $\ell_1$ theorem (...
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Computability of Vector Representations without Inner Products.

Assume you have a separable inner product space $V$ with orthonormal basis $B=(e_a)_{a∈A}$ and an element $v∈V$. Then there is a simple algorithm to compute the representation of $V$ with respect to $...
Hyperplane's user avatar
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Any linear transformation there exists a basis such that $\phi(v_i) = \sum_{j=1}^n a_{ij} v_j$.

Let $V$ be a finite dimensional vector space with basis $\{v_1, \cdots, v_n\}$ over an algebraically closed field $K$. I want to prove the following theorem. For any linear transformation $\phi: V \...
phy_math's user avatar
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Basis for a column space

To find the Col(A) we row reduce the matrix and get $R$ where $R$ is row equivalent to $A$. Now we choose the columns in $A$ corresponding to the pivot columns in $R$ as the basis. My question is when ...
Upstart's user avatar
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Linear Independence Condition

Let $\mathbb{C}^2(\mathbb{R})$ be the vector space of all complex two-tuples over $\mathbb{R}$. I know that the basis is $\{(1,0),(i,0),(0,1),(0,i)\}$. I also know that showing LINEAR DEPENDENCE of ...
Upstart's user avatar
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Uncountable subset $X$ of $\mathbb{R^{n}}$ with property that every subset of $X$ with $n$ elements is a basis of $\mathbb{R}^{n}$.

$\textbf{Question}$: Show that there exist an uncountable subset $X$ of $\mathbb{R}^{n}$ with property that every subset of $X$ with $n$ elements is a basis of $\mathbb{R}^{n}$. $\textbf{My Attempt}$: ...
S Joseph'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{...
5Nik's user avatar
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Is a Hilbert space determined by its algebraic dimension? [duplicate]

Every Hilbert space is determined up to a unitary isomorphism by the cardinality of its orthonormal basis (which exists by Gram-Schmidt). My question: Let $\mathbb{K}$ be the field either $\mathbb{R}$...
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Basis of a polynomial ring over the field of Rationals

I need help checking the validity of the method of my proof and completing it. We observe the polynomial ring $\mathbb{Q}[x] $ as a vector space over the field $\mathbb{Q}$ it is easy to see from the ...
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