Definition of the problem
Let $\mathcal{H}$ be a Hilbert space which consists of functions, defined on a set $S$. Let $k:S\times S \rightarrow \mathbb{K}$ is our reproducing kernel for $\mathcal{H}$. Now, let $\mathcal{H}$ be the two-dimensional subspace of $L^2(0,1)$, consisting of the functions $f(t)=a+bt,\ a,b\in \mathbb{K}$. I am asked to find the reproducing kernel for $\mathcal{H}$.
My efforts
I have to show that $f(t)=\left\langle f,k_{t}\right\rangle $. Given that $f(t)=a+bt$, we have to find $k_t$ such that $a+bt= \left\langle a+bt,k_{t}\right\rangle$. And in $L^2(0,1)$, we know that $\left\langle f,k_{t}\right\rangle =\int f\left(t\right)\overline{k_{t}}dt=f\left(t\right)$.
My question
How could I find from there the reproducing kernel of $\mathcal H$? Should I integrate over $dt$, and between $(0,1)$ ?
Thank you, Franck!