Let $(B_t)_{t\ge 0}$ be a real-valued Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname P)$, $\lambda$ be the Lebesgue measure on $[0,\infty)$ and $$\langle W,\phi\rangle:=\int\phi(t)B_t\;{\rm d}\lambda\;\;\;\text{for }\phi\in\mathcal D:=C_c^\infty([0,\infty))\;.$$
We can prove that the expectation $$\operatorname E[W](\phi):=\operatorname E\left[\langle W,\phi\rangle\right]\;\;\;\text{for }\phi\in\mathcal D$$ is $0$. Now I want to prove that the covariance $$\rho[W](\phi,\psi):=\operatorname E\left[\langle W,\phi\rangle\langle W,\psi\rangle\right]=\int\int\min(s,t)\phi(s)\psi(t)\;{\rm d}\lambda(s)\;{\rm d}\lambda(t)\;\;\;\text{for all }\phi,\psi\in\mathcal D\;.$$
Obviously, we somehow need to use that $$\operatorname{Cov}[B_s,B_t]=\min(s,t)\;\;\;\text{for all }s,t\ge 0\;.$$ How can we do that?
[As a secondary issue: How can we generalize this result to Hilbert space valued Brownian motions and cylindrical Brownian motions?]