For the purpose of practice I took on the following problem about the Riesz representation theorem which asks me to find the unique element whose existence the representation theorem ensures.
Let $H = L^2(0,1)$ and for $u \in H$ let $F$ be the functional
$$ F(u) := \int\limits_{0}^{1/2} u(t)\, dt $$
Show that $F$ is continuous and linear on $H$, calculate its operator norm and find the unique element $f \in L^2(0,1)$ which represents $F$ by Riesz representation theorem
$$ \langle u, f \rangle = \int\limits_{0}^{1} f(t) u(t)\, dt \quad \forall u \in L^2(0,1)$$
The first part I think I solved. The linearity of the functional is quite clear due to the Lebesgue integral properties, i.e.
$$F(\alpha u_1 + u_2) = \int\limits_{0}^{1/2} \alpha u_1 + u_2 dt = \alpha F(u_1) + F(u_2) $$
Also, it's continuous since
$$F(u) = \int\limits_{0}^{1/2} u(t)\, dt \leq \int\limits_{0}^{1} \lvert u \rvert dt \leq \int\limits_{0}^{1} \lvert u(t)\rvert^2\, dt$$
and hence
$$F(u) \leq \Vert u \Vert_{L^{2}(0,1)}$$
What I struggle with is the operator norm and to actually find $f \in L^2(0,1)$. I read that the operator norm of a functional
$$F_g(u) = \int\limits_{\Omega} ug \, dt $$
is given by $\Vert F\Vert = \int\limits_{\Omega} \lvert g\rvert\, dt$. But what is it in my case and how do I find the unique element $f$?
Best Answer
The Riesz representation $f$ of $F$ should obey $$ F(u) = \int fu\,\mathrm dt. $$
Thus it can be seen that $f=\chi_{(0,\frac12)}$ has to be the representant of $F$.
The operator norm for functionals on $L^2(0,1)$ obeys $\| F\|^2 = \int |f|^2 \,\mathrm dt$. This is easy to calculate for the function $f$ above.