Exercise I.A.4

Question

Let $\mathbb{R}^\infty$ be the vector space of all real sequences and let $W\subseteq \mathbb{R}^\infty$ be all sequences with only a finite number of nonzero components. Show that $W$ is a subspace of $\mathbb{R}^\infty$.

Toğrul Topal · Accepted Answer

Obviously $W$ is closed under scalar multiplication, since multiplying a sequence with a scalar does not increase the number of its non-zero elements. If a vector $v\in W$ has $n$ non-zero elements, and the vector $w\in W$ has $m$ non-zero elements, then obviously $v+w$ can have at most $n+m$ non-zero entries.

Exercise I.A.4

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Exercise I.A.4

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