Showing existence of an operator

I've been given the following question:

Let $\displaystyle (\mathcal{H},(,))$ be a real Hilbert space with an orthonormal basis $\displaystyle \left\{e_n\right\}_{n=1}^{\infty} $. Let $\displaystyle F:[0,1]\to \mathcal{H}$ be continuous. Show that there exists a positive self-adjoint operator $\displaystyle T\in B(\mathcal{H})$ such that $\displaystyle (Tx,y) = \int_0^1(F(t),x)(F(t),y)dt$ for all $\displaystyle x,y\in \mathcal{H}$.

Show moreover that $\displaystyle T$ is compact.

(you may use that $\displaystyle \int_0^1\sum_{n=N+1}^{\infty}|(F(t),e_n)|^2dt\to 0$ as $\displaystyle N\to\infty$)

I don't quite understand what must be shown here exactly. Must I find such a operator T explicitly and then show it has the desired property. If so, how? I don't quite see where the hint is useful, I guess it's meant for showing compactness of T.

Any hint or push in the right direction is very much appreciated.