Vector Space of Sequences with Finite Support is Vector Space

Theorem

Let $\struct {K, +, \circ}$ be a division ring.

Let $V$ be the vector space of sequences with finite support in $K$.

Then $V$ is a vector space over $K$.

Proof

Consider $V$ as a subset of the vector space of all mappings from $\N$ to $K$.

Let us apply the One-Step Vector Subspace Test.

Thus, let $\sequence{a_n}, \sequence{b_n} \in V$ be sequences over $K$ with finite support, and let $\lambda \in K$.

Then we need to show that $\sequence{ a_n + \lambda b_n }$ has finite support.

It is immediate that $a_n + \lambda b_n = 0$ if $a_n = b_n = 0$.

Therefore:

$\set{ n \in \N: a_n + \lambda b_n \ne 0} \subseteq \set{ n \in \N: a_n \ne 0 } \cup \set{ n \in \N: b_n \ne 0 }$

By Union of Finite Sets is Finite, it follows that $\sequence{ a_n + \lambda b_n }$ has finite support.

The result follows by the One-Step Vector Subspace Test.

$\blacksquare$