  For spatially dependent functional data, a generalized Karhunen-Lo\`{e}ve
expansion is commonly used to decompose data into an additive form of temporal
components and spatially correlated coefficients. This structure provides a
convenient model to investigate the space-time interactions, but may not hold
for complex spatio-temporal processes. In this work, we introduce the concept
of weak separability, and propose a formal test to examine its validity for
non-replicated spatially stationary functional field. The asymptotic
distribution of the test statistic that adapts to potentially diverging ranks
is derived by constructing lag covariance estimation, which is easy to compute
for practical implementation. We demonstrate the efficacy of the proposed test
via simulations and illustrate its usefulness in two real examples: China
PM$_{2.5}$ data and Harvard Forest data.
