Norms on Functionals 1

**raed** · March 28th 2011, 05:39 AM

Dear Colleagues,

Could you please help me in solving this problem:
Find the norm of the linear functional $f$ defined on $C[-1,1]$ by $f(x)=\int_{-1} ^{0} x(t)dt-\int_{0} ^{1} x(t)dt$ .
I have already proved that $||f||\leq 2$ , it remain to show that $||f||\geq 2$ .
Remark $||x||=max \ x(t), t\in [-1,1]$ .

Regards,

Raed.

**raed** · March 28th 2011, 05:40 AM

Originally Posted by raed

Dear Colleagues,

Could you please help me in solving this problem:
Find the norm of the linear functional $f$ defined on $C[-1,1]$ by $f(x)=\int_{-1} ^{0} x(t)dt-\int_{0} ^{1} x(t)dt$ .
I have already proved that $||f||\leq 2$ , it remain to show that $||f||\geq 2$ .
Remark $||x||=max \ x(t), t\in [-1,1]$ .

Regards,

Raed.

Sorry, $||x||=max \ |x(t)|, t\in [-1,1]$ .

**Opalg** · March 28th 2011, 07:05 AM

Try taking $x(t) = -t^{1/n}$ , 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].

**raed** · March 28th 2011, 09:20 AM

Thank you very much for your reply.

Regards,

Raed.

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