= Cylindrical Wiener process
{c}
A <Cylindrical Wiener process> on a real separable <Hilbert space> $U$ is a linear family of jointly <Gaussian processes> $W_t(h)$, indexed by $h\in U$, with continuous trajectories and
$$
\mathbb E W_t(h)=0,\qquad \mathbb E[W_t(h)W_s(g)]=\min(t,s)\langle h,g\rangle_U.
$$
Linearity can be understood in the space of square-integrable <random variables>. For a unit vector $h$, $W_t(h)$ is standard <Brownian motion>; orthogonal directions give independent <Brownian motions>. The definition specifies all finite collections of observations without requiring a $U$-valued random position.<An Introduction to Stochastic PDEs by Martin Hairer>[Section4.4, Definition4.60 and discussion of identity covariance and larger Hilbert-space realization]{ref}
For an <orthonormal basis> $(e_k)$, let $\beta_k$ be independent standard <Brownian motions>. Then
$$
W_t(h)=\sum_{k=1}^{\infty}\langle h,e_k\rangle\beta_k(t)
$$
converges in mean square for each $h\in U$, since its <variance> is $t\|h\|^2$. The formal vector $\sum_k\beta_k(t)e_k$ is different. Its first $N$ coordinates have expected squared norm $Nt$; for fixed $t>0$, the <strong law of large numbers> gives $N^{-1}\sum_{k=1}^N\beta_k(t)^2\to t$ <almost surely>. The squared norm therefore diverges in infinitely many dimensions. Coordinate observations exist even though the proposed random vector does not belong to $U$.
A <Hilbert-Schmidt operator> $A:U\to H$ repairs this mismatch: $\sum_k\beta_k(t)Ae_k$ converges as an $H$-valued square-integrable <random variable> when $\sum_k\|Ae_k\|_H^2$ is finite.<An Introduction to Stochastic PDEs by Martin Hairer>[Section4.4 after Definition4.60; Theorem4.47 applied to Hilbert-Schmidt maps]{ref} For example, on $\ell^2$, setting $Ae_k=k^{-1}e_k$ gives expected squared norm $t\sum_k k^{-2}=t\pi^2/6$. This smoothing makes an actual random position, while retaining infinitely many noisy coordinates. It is useful when defining <stochastic partial differential equations> and their noise terms.
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