Question

Problem 4. We denote by an operator f on L²(0,1) f(u) = 'u(s)ds, ue Lº(0,1). (1) Prove that f € (L(0,1))* (2) There exists v € L? (0,1) such that f(u) = (u, v)L2(0,1) for any u € L?(0,1). Then, write v € L? (0,1) explicitly.

Answer 1

1) Note that since $u \in L^2((0,1))$ , therefore u is bounded in (0,1). Therefore, the operator f is also bounded. Now consider

$\\ \forall u,v \in L^2((0,1)), c\in \mathbb C, f(cu+v) = \int_0^1 cu(s)+v(s) ds = c\int_0^1u(s)ds+\int_0^1v(s)ds = cf(u)+f(v)$

Hence, we can conclude that f is an linear bounded operator, thus $f \in (L^2(0,1))^*$

2) Note that if we take v(s) = 1 for all s in (0,1), then we get that

$(u,v) = \int_0^1 u(s)\overline{v(s)}ds = \int_0^1 u(s)ds = f(u) \forall u \in L^2(0,1)$

Now, even if there are discontinuities to 0, it will still be integrable. Therefore, we get that the formula for v is

$v(s) = 1, \forall s \in (0,1)\setminus\mathcal A, |\mathcal A| = |\mathbb N|$