Question

Linear Algebra homework help please
A1)Consider the set of convergent sequences ,with the same addition and scalar multiplication that we defined for the space of sequences:
V={f|f:ℕ->ℝ,lim(n->∞) f(n) ∈ ℝ }⊂ ℝ^ℕ
Is this a vector space?Explain why or why not.

1b)now consider the set of divergent sequences, with the same addition and scalar multiplication as before:
V={f|f:ℕ->ℝ,lim(n->∞) f(n) doesnt exist or (+-)∞ }⊂ ℝ^ℕ
Is this a vector space?Explain why or why not.

Answer 1

What is the definition for a vector space for functions?

Answer 2

heres some notes

Answer 3

So I'm guessing we need to show that addition and scalar multiplication are closed, or not closed, on these vector spaces.

Answer 4

For the first one, we'd ask:
Take two convergent functions, add them, is the result necessarily convergent?
Take a convergent functions and multiply it by a real number, is the result necessarily convergent?

Answer 5

thank you=)