Confused on the definitions of norm of a function.

Question

If $$\|f \| =\sup \{|f(x)|:x \in [0,1]\} $$ and also $$ \|f \|=\int^1_0 |f(x)| \, dx,$$ then for $f(x)=x$, we have $\sup \{|f(x)|:x \in [0,1]\} = 1$. But $\int^1_0 |f (x)| \, dx = \int^1_0 |x| \, dx= \frac{1}{2}$. Then how is it possible ?

Answer 1

A vector space can be associated with different norms, all satisfying norm axioms. In your example, the first is $L^\infty$ norm and the second $L^1$ norm.