I'm working with operators and I have two questions about their norm.
1) Usually, to find a norm I apply the operator to a chosen function and I basically see what happens. Provided that the test function needs to belong to the initial space, how do I "cleverly" choose one?
2) I also found this : let $T:V\rightarrow W$ and consider the domain of the operator, which will obviously be $D_t \subset V$ $$||T||=sup_{x \in D_t, x\not=0} {||Tf||_W \over ||f||_V}$$
But how do I recognise the domain of an operator? Is it simply where it operates? i.e. (few examples):
1) $T:L^1(R)\rightarrow L^1(R)$, $Tf=\int_Re^xf(x)dx \rightarrow D_T=R$
2) $T:L^1(R)\rightarrow L^1(R)$, $Tf=\int_0^1e^xf(x)dx \rightarrow D_T=[0,1]$
3) $T:L^2(0,2)\rightarrow L^2(0,2)$, $Tf= {f(x) \over 1-x}\rightarrow D_T=(0,2)$ excluding $1$?
Thanks!