= Cylindrical Brownian motion
{title2=$\mathbb E[W_t(h)W_s(k)]=(t\wedge s)\langle h,k\rangle$}
On a real <Hilbert space> $H$, this centered jointly <Gaussian process> is linear in its test vector $h$ and has the displayed covariance, with independent increments in time. Equivalently choose an orthonormal basis and independent real <Brownian motions> $W^j$, and define stochastic integration by $\int\langle v_s,dW_s\rangle=\sum_j\int\langle v_s,e_j\rangle dW_s^j$ whenever $\int\|v_s\|^2ds<\infty$. In infinite dimension it need not be an $H$-valued <random variable>; the integral is the meaningful object.
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