# 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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### 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'...
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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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### 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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### 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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### 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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### 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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