  In this note we present the metric approximation property for weighted
mixed-norm $L_w^{(p_1,\dots ,p_n)}$ and variable exponent Lebesgue type spaces.
As a consequence, this also implies the same property for modulation and
Wiener-Amalgam spaces. We then characterise nuclear operators on such spaces
and state the corresponding Grothendieck-Lidskii trace formulae. We apply the
obtained results to derive criteria for nuclearity and trace formulae for
periodic operators on $\mathbb R^n$ and functions of the harmonic oscillator in
terms of the global symbol.
