Suppose:
$$
dX\_t = a(t,X\_t)dt + b(t,X\_t)dW^H\_t
$$
is an SDE with values in a separable Hilbert Space $H$, and $W^H\_t$ is an $H$-valued cylindrical Wiener process. Then can we write the dynamics for $X\_t$ in terms of a basis $\{e\_i\}$ of $H$?


That is, if $\{e\_i\}$ is a basis for $H$ then is the previous SDE equivalent to:
$$
dX\_t= \sum\_i dX\_te\_i =\sum\_i \left(a(t,X\_t)e\_idt + b(t,X\_t)e\_idW\_t^H \right)
?
$$



Hilbert-Space Values SDE in terms of Basis
In principle, the *variational form* of the SPDE problem allows you to do this. What you do is expand the solution in terms of basis vectors, and choose the test functions in the variational formulation to be basis vectors. This is how projection-based methods for SPDEs are constructed. 


For more detail see, e.g., Definition 10.19 and Section 10.6 of: 
**An Introduction to Computational Stochastic PDEs** Part of Cambridge Texts in Applied Mathematics by Gabriel J. Lord, Catherine E. Powell, Tony Shardlow


