This is one of the problems from the book: Hoffman and Kunze, chapter: Vector Spaces
Let $V$ be the (real) vector space of all functions $f$ from $\mathbb{R}$ into $\mathbb{R}$. Is the set of all $f$ which are continuous, subspace of $V$?
I am not sure how to proceed with the solution.
Best Answer
You should apply the subspace test. Suppose $V$ is the vector space of all functions from $\mathbb{R}$ to $\mathbb{R}$, and $W$ is the subset of all continuous functions in $V$. To see if $W$ is a subspace of $V$ it's enough to check that $W$ is non-empty, and that it's closed under vector addition and scalar multiplication.
It's clear that $W$ is non-empty, since, for example, it contains the constant zero function.
Showing that $W$ is closed under scalar multiplication means checking that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous, and $c\in \mathbb{R}$ is a real number, then the function $c.f : x \mapsto cf(x)$ is also continuous. Showing that $W$ is closed under vector addition means checking that if $f$ and $g$ are continuous functions (belonging to $W$), then so is $f+g: x \mapsto f(x)+g(x)$.
The essential facts needed to prove these two properties are the following: the addition and multiplication functions $+: \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}$ and $.: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ are both continuous. If $f$ and $g$ are two continuous functions from $\mathbb{R}$ to $\mathbb{R}$, then the composition function $f\circ g: x \mapsto f(g(x))$ is also continuous.