Let $T:C^0(\mathbb{R},\mathbb{R}) \rightarrow C^0(\mathbb{R},\mathbb{R})$ defined by $T(f)(x)=\displaystyle \int_0^x f(t)dt$ . If $T^*: C^0(\mathbb{R},\mathbb{R})^* \rightarrow C^0(\mathbb{R},\mathbb{R})^*$ indicates the Transpose of a Linear Transformation, how find an element non-zero (i.e. a function non-zero) in $Kern (T^*)$?
I was thinking of defining $f=L_0$, because $T^*(f)(\alpha)=T^*(L_0)(\alpha)=(L_0)(T\alpha)=T\alpha(v)= \int_0^0 f(t)dt=0$, but $L_0$ is always null, this is the problem. Any better ideas?
The dual space is the space of Borel measures on $\mathbb R$. The measure $\delta_0$ defined by $\delta_0 (A)=1$ if $0 \in A$ and $0$ otherwise is a non-zero element in the kernel of $T^{*}$. [$T^{*}\delta_0 (f)=\delta_0 (Tf) =Tf(0)=0$ for all $f$].