What is the Dimension and Cardinality of the Vector Space V?

[Chris L T521]

Here's this week's problem!

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Problem: Let $\mathbb{F}_2$ denote the field with two elements (called $0$ and $1$). Let $V=\mathbb{F}_2\mathbb{N}$ denote the vector space whose elements are sequences $(a_i)_{i\in\mathbb{N}}$, such that $a_i=0$ for all but finitely many $i\in\mathbb{N}$. What is $\dim(V)$? What is $\#V$? What is $\#V^{\prime}$? What is $\dim(V^{\prime})$?

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[Chris L T521]

No one answered this week's problem. You can find my solution below.

[sp]Consider the set$$\mathcal{B}=\{(a_i)_{i\in\mathbb{N}}: 1\text{ is in the $i$th position and $0$ otherwise}\}.$$
This forms a basis for the free vector space $V=\mathbb{F}_2\mathbb{N}$. Therefore,
$$\dim(V)=\#\mathcal{B}=\#\mathbb{N}=\aleph_0.$$
Since $\#\mathcal{B}$ is infinite, then it follows that $\#V=\max(\#\mathbb{F}_2,\#\mathcal{B})=\aleph_0$.
Since $\dim(V)$ is infinite, then it follows that $\dim(V^{\prime})=\left|\mathbb{F}_2\right|^{\dim(V)}=2^{\aleph_0}=\mathfrak{c}.$ It also follows that $\#V^{\prime}=\max(\#\mathbb{F}_2,\dim(V^{\prime}))=\mathfrak{c}.\quad\clubsuit$[/sp]