I've been working on algebra and want to know how to determine if a certain type of function is a subspace of the vector space $\mathbb{R} \to \mathbb{R}$. So far I've been using the two properties of a subspace given in class when proving these sorts of questions, $$\forall w_1, w_2 \in W \Rightarrow w_1 + w_2 \in W$$ and $$\forall \alpha \in \mathbb{F}, w \in W \Rightarrow \alpha w \in W$$ The types of functions to show whether they are a subspace or not are:
(1) Functions with value $0$ on a specified set $S\subset \mathbb{R}$
(2) Functions with only finitely many discontinuity points
(3) Functions with value $0$ outisde of a finite set
I mainly face difficulty generalizing these type of functions into some form say $f(a)=b$ as to have a $w_1, w_2$ to manipulate and prove whether or not it is a subspace using the properties of subspace.
Best Answer
(1) Suppose $f(x)=g(x) = 0$ for all $x \in S$, then is the same true of $\lambda f$ and $f+g$?
(2) Suppose $f,g$ have only finitely many (possibly different) points of discontinuity. What about $\lambda f$ and $f+g$? Do they have finitely many points of discontinuity?
(3) Suppose $f,g$ are zero for all except finitely many (possibly different for $f,g$) points. Is the same true of $\lambda f$ and $f+g$?