Define $\|\cdot\| \colon V \rightarrow \mathbb{R} $ by $\|a\| =\sum 2^n |a_n| $, then choose the correct option

Question

Define $\|\cdot\|\colon V \rightarrow \mathbb{R} $ by $\|a\| = \sum 2^n |a_n| $, where $V$ denotes the vector space of all sequences $a=(a_1,a_2,a_3,\dotsc)$ of real numbers such that $\sum 2^n |a_n| $ converges.

Which of the following are true?

1) $V$ contains only the sequence $(0,0,\dotsc)$

2) $V$ is finite dimensional

3) $V$ has a countable linear basis

4) $V$ is a complete normed space

My answer: it is already given that $\sum 2^n |a_n| $ converges. That means $V$ contain contains only the sequence $(0,0,\dotsc)$ so option 1) is correct.

Option 2 is incorrect because $\mathbb{R}$ is uncountable, as continuous image of uncountable is uncountable.

I'm confused about option 3 and option 4.

Please help me.

Any hints/solution will be appreciated.

Answer 1

Option 1 is incorrect, because all sequences having only finitely many nonzero entries belong to $V$. This also easily shows that 2 is incorrect.

Also option 3 is incorrect, because a complete normed vector space cannot have a countably infinite basis (either finite or uncountable).

On the other hand, option 4 is correct, because…

Answer 2

Hint:

The map $\ell^1 \to V$ given by $$(x_n)_n \mapsto \left(\frac{x_n}{2^n}\right)_n$$ is an isometric isomorphism.

Therefore just check what properties hold for $\ell^1$ (only option $4$).