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Qc) Suppose f(t) belongs to V. For a complex number c, show that h(t) =cf(t) also belongs to V.

Answer

Letfbe a periodic function of period 1 andh =cfwhere c is a complex number.

∀t h(t+1) =cf(t+1) =cf(t) = h(t)

Hencehis periodic with period 1 QED.

Yes, that's it.

Qd)

Is V with the above addition and scaling operations a vector space?

Would I need to go through the axioms to prove that V was definitely a vector space?

It depends. If you can assume that the set of all complex functions is a vector space, then you need only show that addition and scalar multiplication are closed on the subset you are dealing with (in this case functions with period 1).

This is true for vector spaces in general:

If ##U## is a subset of a vector space ##V## and:

##u, v \in U## and ##c \in \mathbb{C} \ \Rightarrow u + v \in U## and ##cu \in U##

then ##U## is a vector subspace. Most axioms are met directly because the axioms hold for ##V##, but you might like to think about why the above ensures that the ##0## vector is in ##U## and also why it ensures that ##u \in U \Rightarrow -u \in U##. And, you might like to look at the other axioms as well and convince yourself why they must be met for any subset of a vector space.

With that assumption, you have shown that the set of functions of period 1 is a vector space.