Define an "index space" to be any topological space $I$, which is separable and locally compact Hausdorff. Define a "value space" to be any topological vector space $V$, which is separable, locally convex and complete.

Define a $V$-valued array (indexed by $I$) to be a continuous function $a : I \to V$. The space of such arrays is $V^I := C(I,V)$, equipped with the compact-open topology. An array space is also a value space, so it makes sense to consider arrays-of-arrays.

Define the curry operator $\operatorname{curry} : V^{I \times J} \to (V^J)^I$ by $\operatorname{curry}(a)(i)(j) := a(i,j)$. Is the curry operator a homeomorphism of array spaces? i.e., is it the case that $V^{I \times J} \cong (V^J)^I$?