A simple problem about adjoint operators

Question

Consider the operator $T:L^2(0,1)\rightarrow C([0,1])$ defined by

$$Tx(t) := \int_0^tx(s)ds.$$

Pick any $a \in [0,1] $ and let be $\delta_a$ be the Dirac delta at $a$. If $T'$ is the operator adjoint of $T$, then I am supposed to find some $g \in L^\infty (0, 1)$ which defines $T'\delta_a$. I do not really understand the question, how could some $g \in L^\infty (0, 1)$ define $T'\delta_a$? Please do not give me the function $g$ since this is an exercise that I will be graded on, I just want to understand the question. How should I think here to come up with $g$?

Answer 1

First note that

$$<\delta_a, Tx> = \int_{0}^{a} x(s)ds$$

So is there a $g := T'\delta_a$ such that

$$<g, x> = \int_{0}^{1} g(t)x(t) dt = \int_{0}^{a} x(t)dt$$

??

Answer

$$ g(t) = 1 $$ for $t < a$ and $$ g(t) = 0 $$ for $t > a$

Answer 2

HINT: just find such a $g$ in $L^2(0,1)$. Once you get it, you can check that it is also an element of $L^{\infty}(0,1)$