# Thread: Norms on Functionals 1

1. ## Norms on Functionals 1

Dear Colleagues,

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

Regards,

Raed.

2. Originally Posted by raed
Dear Colleagues,

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

Regards,

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

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