what is difference between 0 and infinity norm? Suppose  $f$ is a real function on $\Omega$, both $\|f\|_\infty$ and $\|f\|_0$ are defined as
$\sup_{x\in \Omega} f(x)$ in many books. Then, am I missing some from their definitions?
 A: In the context that these are functions on a compact topological space $X$, the $C^o(X)$ norm is the sup norm, which, in the context of Riesz and others, is a limit of $L^p$-norms, so, called the $L^\infty$-norm.
For non-compact $X$, the continuous functions usually need not have finite sups, etc, so there'd be a divergence of notation and concept.
That is, obviously abbreviations and economy are desirable and useful, and/but the specific choices are heavily dependent on context: not only when, but "which demographic". 
A: I can't speak for the books that you've read, but when I've seen it, the $0$-norm is defined differently.  Some books define $\|f\|_0$ as $|\{x|f(x)\neq 0\}$, that is, the measure of the support of $f$.  Equivalently, for finite-dimensional vector spaces, $\|f\|_0$ is the number of non-zero elements.  Strictly speaking, this is an abuse of terminology since the "$0$-norm" is not really a norm since it does not have the property of scalability.  However, it can be said to arise as the limiting case of the $L^p$-norm as $p\to0$.
