- #1

iJake

- 41

- 0

## Homework Statement

Let ##V## be the vector space of the sequences which take real values. Prove whether or not the following subsets ##W \in V## are subspaces of ##(V, +, \cdot)##

a) ## W = \{(a_n) \in V : \sum_{n=1}^\infty |a_n| < \infty\} ##

b) ## W = \{(a_n) \in V : \lim_{n\to \infty} a_n = 1\}##

c) ## W = \{(a_n) \in V : ∃ \lim_{n\to \infty} a_2n\}##

d) ## W = \{(a_n) \in V : (cardinality) \{n : a_n \neq 0\} < \infty \}##

The # for my cardinality in the last subspace was breaking my formatting.

## Homework Equations

The three conditions to be a subspace under ##V## are that vector addition and scalar multiplication be closed, and that the 0 vector belongs to the subspace.

## The Attempt at a Solution

[/B]

The notation and logic here is throwing me off a bit because I have not yet begun to study Calculus and I am unfamiliar with the operations and behavior of sequences.

For a)

##a_{n+2} + b_{n+2} = (a_{n+1} + b_{n+1}) + (a_n + b_n)##

##\lambda \cdot (a_n, a_{n+1}, a_{n+2}) = (\lambda a_n, \lambda a_{n+1}) \lambda a_{n+2}) \forall \lambda \in \mathbb k , \forall a_n \in W ##

if ##a_n = 0##, the sequence is 0, and ##b_n + 0 = b_n##

##0 \cdot a_n = 0 \forall a_n \in W##

I conclude that in a) W is a subspace.

In b), I recognize that there is a problem with the 0 vector, but I'm not sure how to describe it in notation. I don't think that b) is a subspace.

For c) I think that all of the properties should hold, but I am again confused about how to write out the notation. For example, to prove closure under addition, should I be adding limits? For closure under multiplication, multiplying scalars by limits? Sorry if this is a bit of a silly question, but I just don't know how to write it out. I feel like all of the properties should hold, however.

In d) there is again a problem with the 0 vector, in that ##a_n = 0 ∉ W##, thus I conclude d) is not a subspace.

All help is greatly appreciated.