Extend $T$ to a bounded linear operator...

Question

Let $X$ be the real inner product space of real-valued continuous functions on $[0,2\pi]$ with the standard inner product. Consider the map $T:X \to H$, where $H=L^2[0,2\pi]$ (the completion of $X$), defined by

$$Tx = \int_0^{2\pi}cos(t-s)x(s)ds.$$

Also, keep in mind that $(1/\sqrt{2\pi},\cos(t)/\sqrt{\pi},\sin(t)/\sqrt{\pi},\cos(2t)/\sqrt{\pi},\sin(2t)/\sqrt{\pi},\dots)$ is a total orthonormal sequence in $H$.

1. Prove that $T$ is a bounded linear operator.
2. Extend $T$ to a bounded linear operator $T^*:H \to H$.

I have already shown part (1). I am not sure how to proceed with part (2). I know there are Theorems that say that we can extend bounded linear operators, but I am not seeing how to use them.