I'm reading a paper recently and have come across with the concept of duality pairing and test function. In particular, I met a function $Z$ on $\mathbb{R}\times \mathbb{R}^d,$ which is the solution of the stochastic heat equation given by:
$$Z(t,x)=\int_{\Lambda_t} K(t-s,x-y)\xi(s,y)dyds, $$
where $K$ is the heat kernel and $\xi$ is the so-called white noise.
The paper said testing $Z$ against a test function $\eta:\mathbb{R}^{d+1}\to \mathbb{R}$ yields the duality pairing:
$$(Z,\eta)=\int_\Lambda \int_\Lambda \eta(z_1)K(z_1-z_2)dz_1\xi(dz_2), $$
and in the above integral we have combined the space and time variable in to a single variable $z=(t,x).$
However, since I haven't studied functional analysis I couldn't quite understand where does the expression of $(Z,\eta)$ come from, could anyone provide with me some calculation to get this, or some reference or textbook to read?
After some self-study, my naive understanding is that $(Z,\eta)$ should equal to $Z(\eta),$ but since $\eta$ takes value on $\mathbb{R},$ is seems to be very wrong here?
Thank you very much for any of your help in advance:)!