norm of the functional $T(f)=\int_0^2f(x)dx-\int_3^{10}f(x)dx $ on $ L^2[0,10]$

Question

Let

$L^2[0,10]=\{f:[0,10]\rightarrow R | f $is Lebesgue measurable and $\int_0^{10} f^2(x)dx \lt \infty \} $ eqipped with the norm $|| f ||=\big(\int_0^{10} f^2(x)dx \big)^{1/2} $ and $T$ be the linear functional on $L^2[0,10]$ given by

$T(f)=\int_0^2f(x)dx-\int_3^{10}f(x)dx $

Then $|| T || $=_________

I have no idea how to approach this problem .Where to use the fact that $f$ is Lebesgue measurable? Please help me with me useful hints. Thanks for your time.

Answer 1

Suggestion:

Find a function $g(x)\in L^2$ such that your functional has the form $\langle f,g\rangle$ (the usual inner product). There must be such a function by the Riesz representation theorem. The norm of the functional is then just the norm of $g$ in $L^2$.