Dear Colleagues,
Could you please help me in solving this problem:
Find the norm of the linear functional defined on by .
I have already proved that , it remain to show that .
Remark .
Regards,
Raed.
Try taking , where n is an odd integer (so that x(t) is defined when t is negative).
The idea is that you want x(t) to be close to 1 when t is positive, and close to +1 when t is negative. But x has to be a continuous function, so it will have to change rapidly as t goes from negative to positive. Another choice for x(t) would be to define it to be +1 in the interval [1,1/n], 1 in the interval [1/n,1], and x(t) = nt in the interval [1/n,1/n].