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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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 ...
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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 ...
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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,...
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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 ...
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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 ...
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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 $\...
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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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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 ...
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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 $...
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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$...
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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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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 ...
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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 ...
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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 ...
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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}...
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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 ...
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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. ...
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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 $...
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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 \...
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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 ...
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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 ...
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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}$: ...
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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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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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Boundness of the operator when Hamel basis vectors are eigenvectors
Let $X$ be infinite-dimensional Banach space, $\{e_j\}_{j\in J}$ be normalized Hamel basis for it.
Suppose for each $j\in J$ we choose a number $\lambda_j\in\mathbb{R}$. Then we define a linear map
$$...
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Largest and smallest norm on a vector space given a Hamel basis should be normalised in that norm
I don't know whether the answer to this question will help answering this question.
Let $V$ be an infinite-dimensional vector space and $A \subset V$ a Hamel basis in $V$. What is the largest (and ...
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Why isn't this a counterexample of Banach-Steinhaus theorem?
The theorem from Wikipedia is as follows
Let $X$ be Banach space, $Y$ be a normed space and $F$ be family of linear bounded operators $f:X \to Y$ such that $\forall x \in X \sup_{f \in F} \|f(x)\|_Y &...
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What is a Hamel basis?
According to Mathworld, a Hamel basis is a basis for $\mathbb R$ considered as a vector space over $\mathbb Q$.
According to Wikipedia, the term is used in the context of infinite-dimensional vector ...
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How to Find the Basis for a Specific Vector Space? [closed]
I have the following homework problem:
In Exercises 1-4, $W$ is a subspace of the vector space $V$
of all $(2 \times 2)$ matrices. A matrix $A$ in $W$ is written as
A =
$$
\begin{bmatrix}
a & b \\...
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Basis for a subspace of matrix $2\times2 $with sum 0 entries.
Consider the subspace of matrices of the form $\left(\begin{matrix} a & b\\ c & d \end{matrix}\right)$ such that $a+b+c+d=0$ I have to find a basis. I was thinking for example of letting $3$ ...
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What am I missing in this paper: ``Infinite dimensional Banach spaces must have uncountable basis—an elementary proof"?
In the paper (or refer to this answer to view the full proof of this result):
Tsing N.K. [1984]. Infinite dimensional Banach spaces must have uncountable basis—an elementary proof. Amer. Math. ...
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Proving $d\varphi^{\mu_1} \wedge d\varphi^{\mu_2}... \wedge d\varphi^{\mu_k}$ is basis for space of $k$-forms
Let $X$ be a smooth $n$-manifold with a local chart $(U,\varphi)$. Taking the wedge product of the chart induced covariant basis $\{d\varphi^\mu\}$:
\begin{align}
\{d\varphi^{\mu_1} \wedge d\varphi^{\...
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setting up for basis of vector
I don't know how to get started with this
Find a basis for the subspace $V$ of $\mathbb{R}^4$ consisting of all vectors of the form $(a,b,c,a+b+c)$, and state its dimension. Another way or writing ...
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