I'm learning about functions of bounded variations and need to verify my work for this problem:
Show that $\| f \|_{BV} = | f(a) | + V_{a}^{b} f$ defines a norm in the space $BV[a,b]$.
My attempt and Thoughts:
First. We want to show that $\| f \|_{BV} \geq 0$ (the norm is always positive).
This is the same as writing that $| f(a) | + V_{a}^{b} f \geq 0$ (by the definition of $\| f \|_{BV}$).
Well, clearly $| f(a) | \geq 0$ and since $V_{a}^{b} f$ is defined as
$$V_{a}^{b} f = \sup_{P} V(f, P)$$ and we know that $V(f, P) \geq 0$. Hence, $\| f \|_{BV} \geq 0$.
Second. We want to show that $\| f \|_{BV} = 0 \iff f = 0$.
"$\impliedby$": if $f = 0$ i.e. $f(x) = 0, \forall{x} \in [a,b]$ then $\| f \|_{BV} = 0$.
"$\implies$": if $\| f \|_{BV} = 0$, then $| f(a) | + V_{a}^{b} f = 0$ if and only if $f$ is constant.
Third. We want to show that $\|\alpha \ f \|_{BV} = | \alpha | \| f \|_{BV}$.
We have $\|\alpha \ f \|_{BV} = |\alpha \ f(a) | + V_{a}^{b} (\alpha \ f) = | \alpha | | f(a) | + | \alpha | V_{a}^{b} f = | \alpha |(| f(a) | + V_{a}^{b} f) = | \alpha | \| f \|_{BV}$.
Fourth. We must prove that $\| f + g \|_{BV} \leq \| f \|_{BV} + \| g \|_{BV}$.
We have that $\| f + g \|_{BV} = | f(a) + g(a) | + V_{a}^{b} (f + g) \leq | f(a) | + |g(a) | + V_{a}^{b} f + V_{a}^{b} g = \| f \|_{BV} + \| g \|_{BV}$.
Since I'm new to this subject I would like to verify that my work is correct and know if there are parts of the demonstration that I can improve.