# 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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### Deriving a basis from a generating subset and a free subset

Let $K$ be a field and $E$ a $K$-vector space. Let $S$ be a generating system $E$ and $L$ a free subset of $E$ such that $L\subset S$. Then there exists a basis of $E$ such that $L\subset B\subset S$. ...
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### Show linear independence of sum of vectors from a base

let $V$ be a vectorspace and $\{a_1,a_2,a_3,a_4\}$ a base of $V$; Moreover $b_1:=2a_1-a_2,$ $b_2:=a_2+a_3+a_4,$ $b_3:=a_3-a_4$; Show that $b_1,b_2,b_3$ are linear independent; By definition of linear ...
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### Importance of Hamel bases for description of additive functions

I am unable to understand the reason and motivation of the author that why did he set $$f ( x ) = \sum r _ \alpha s ( h _ \alpha ) \text .$$ The importance of Hamel bases for description of ...
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### What is the dimension of the vector space consisting of all real-valued functions?

The dimension of this vector space is obviously infinite dimensional, and it's not too much work to show that its basis is an uncountable set, making it an uncountably-infinite dimensional vector ...
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### Why from the point of view of constructive analysis denumerable hamel basis is not suitable for an infinite dimensional Banach Space

We know that for an infinite dimensional Banach Space the Hamel Basis needs to be uncountable. Why is it difficult from the point of view of constructive analysis to obtain an Uncountable Hamel Basis ...
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### Taft-Hopf Algebra has dimension $N^2$?

Definition of the Taft-Hopf Algebra Let $k$ be a field. Let be $N$ a positive integer such that there exists a primitive $N$-th root of unitiy $\zeta$ over $k$. Denote by $(H, \mu, 1_H)$ the unital, ...
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### Another approach for dimension of a vector space

I am preparing a lecture note for a primary course on Vector Spaces and I am developing basis and linearly independent sets. There I took the following path: Define the linear span $L(S)$ of a subset ...
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### Uncountable Hamel basis of Banach space

I came across this problem: Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.. The proof is absolutely correct however I was wondering if this is true ...
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### What's the dimension of $K^S$ for arbitrary $S$?

Let $K$ be a field, $S$ an arbitrary set, and $K^S$ denote the vector space of functions from $S$ to $K$. What is the dimension of this space? By dimension, I mean the cardinality of a Hamel basis. ...
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### Show that a linear functional $f$ on a vector space $X$ is uniquely determined b its values on a Hamel basis for $X.$ [closed]

Show that a linear functional $f$ on a vector space $X$ is uniquely determined by its values on a Hamel basis for $X$. I need to prove this.
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Here is the first question: Let $X$ and $Y \neq \{0\}$ be normed spaces, where $dim X = \infty.$ Show that there is at least one unbounded linear operator $T: X \rightarrow Y.$ (Use a Hamel basis.) ...
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### Definition for Basis of a Subspace

As I gradually work through the fundamentals of linear algebra, I have often found myself struggling with boiling down the concepts that I have learned to a concise, all encompassing definition which ...
Proposition: Let $(V,F)$ and $(W,F)$ be non-zero vector spaces, and $T$ be a surjective linear map of $V$ onto $W$. Assume that property $(1)$ holds: for any subset S of V, we have that "TS ...