properties of a norm I am quite bothered by this problem in Royden's 4th ed. Show that $||f||=\int_{a}^{b} {x^2 |f|}$ is a norm given that $f\in L^1$ on $[a,b]$.
I want to show that $||f||=0$ iff $f=0$.
Am I doing this right?
if suppose $||f||=0$ then $||f||=\int_{a}^{b} {x^2 |f|}=0$
that is, $0=||f||=|f| \int_{a}^{b} {x^2 }$
then $|f|=0$ thus $f=0$.
am i right to assume that $|f|$ is a constant?

The triangle inequality:
$||f+g|| = \int_a^b x^2 |f(x)+g(x)|dx \leq \int_a^b x^2 (|f(x)| + |g(x)|)= \int_a^b x^2 |f(x) + \int_a^b x^2 |g(x)| = ||f|| + ||g||$
Therefore
$||f+g|| \leq ||f|| + ||g||$
Positive Homogeneity:
$||\alpha f||= \int_a^b x^2 |\alpha f(x)| dx =|\alpha| \int_a^b x^2 |f(x)| =|\alpha| ||f||$
Therefore $||\alpha f || = |\alpha| ||f||$
Nonnegativity: Trivial
Therefore $||f||=\int\limits_a^b x^2 |f(x)| dx$ is a norm.
