Show that this space is a vector space I am looking through a paper that says:
If for the space $L_p(a,b)$, $\int_a^{b} |x(t)|^p dt$ exists, this space is a vector space.
I would like to convince myself that this is true, but I have no idea why. In an undergrad course on linear algebra, I usually used to prove that something was a vector space by showing that all axioms were satisfied. I don't know if/how that can be done here.
Any help?
 A: The reason is a little bit tricky given no background, and you would see this at the beginning of a course in so-called functional analysis when the $L_p$ spaces are introduced. The result is implied by something called Hölder's inequality, which says that
$$
\int_a^b \lvert f(x) g(x) \rvert d x
\leq
\left(\int_a^b \lvert f(x) \rvert^p d x\right)^\frac{1}{p} 
\left(\int_a^b \lvert g(x) \rvert^q d x\right)^\frac{1}{q} 
$$
for functions $f(x)$ and $g(x)$ and $p, q \in (1, \infty)$ satisfying $\frac{1}{p} + \frac{1}{q} = 1$. This in turn can be used to prove Minkowski's inequality, which says
$$
\left(\int_a^b \lvert f(x) + g(x) \rvert^p d x\right)^\frac{1}{p}
\leq
\left(\int_a^b \lvert f(x) \rvert^p d x\right)^\frac{1}{p}
+
\left(\int_a^b \lvert g(x) \rvert^p d x\right)^\frac{1}{p}
$$
for $p \in (1, \infty)$. The fact that $L_p([a,b])$---the space of functions $f(x)$ for which $\int_a^b \lvert f(x) \rvert^p d x < \infty$---is a vector space now follows immediately from this last inequality.
(Note that here by a "function $f(x)$" we mean something called a measurable function.)
