In Stephen Boyd's book, Boyd uses the theorem that a linear function is bounded below on $R^m$ only when it is zero. I can't really digest this. Can someone tell me why this holds?
I mean if I take a line, convert it into a ray. It can start at any point. So indeed its bounded below and its linear but it doesn't mean that it's zero.
Please correct if I am wrong somewhere.
Best Answer
Suppose $f$ is linear, and $f(x) \geq B$ for all $x$. Choose any $x_0$. Then $f(\lambda x_0) = \lambda f(x_0) \geq B$ for all $\lambda$.
Take $\lambda >0$, which gives $f(x_0) \geq \frac{B}{\lambda}$, and since $\lambda$ is arbitrary, we have $f(x_0) \geq 0$.
Now take $\lambda <0$, which gives $f(x_0) \leq \frac{B}{\lambda}$, which now results in $f(x_0) \leq 0$.
Together, this gives $f(x_0) = 0$. Since $x_0$ was arbitrary, we have $f=0$.