In introductions to abstract Wiener spaces, the sample paths usually form a Banach space; so, in particular, the sum of two sample paths is a valid sample path and also an element of the Banach space. Then the Gaussian measure is constructed as a measure on this Banach space, the Cameron–Martin space is a subspace thereof, etc.

However, when the stochastic process is pinned to some non-zero value, then the addition of two sample paths no longer yields a valid sample path, hence the sample paths do not form a Banach space and the whole construction breaks down—so how are abstract Wiener spaces constructed for such pinned processes?

A simple example of this is a Brownian bridge, which is typically constructed to be pinned to zero at both time 0 and some time $T > 0$, but what if it is pinned to some value $b \neq 0$ at time $T$? Is the only way out to take the zero-pinned Wiener space and add $b\tau/T$ to every sample path? So then in a sense there is no true Wiener space for non-zero-pinned processes (since this additional term takes it out of the Banach space)? Also, how do I know that a linear drift is correct instead of some other drift function?

Note: I substantially modified this question after realizing that I had originally kind of missed the point.