fa.functional analysis - Are smooth functions on an uncountable sum continuous?

Not all smooth functions are continuous. It is a fact of the Frölicher−Kriegl−Michor theory that bounded multilinear maps are smooth. For example the canonical bilinear evaluation $Etimes E'tomathbb R$ given by $(x,u)mapsto u(x)$ is bounded, hence smooth, but discontinuous when $E=sum_{mathbb R}mathbb R$. I too quickly thought that this would give the required smooth discontinuous map as a composite $Eto Etimes Eto Etimes E'tomathbb R$.

Using Jarchow's notation, and istead considering the space $F=mathbb R^{ mathbb N}timesmathbb R^{ (mathbb N)}=prod_{mathbb N}mathbb Rtimessum_{mathbb N}mathbb R$ , then one has the Frölicher−Kriegl smooth discontinuous map $Ftomathbb R$ given by $(x,y)mapstosum_{iinmathbb N}(x_icdot y_i)$ .

It should be noted that this discontinuity is with respect to the locally convex topology. Frölicher−Kriegl smooth maps are always continuous with respect to the Mackey−closure topology whose open sets are precisely the $Usubseteq F$ such that for every $xin U$ and every bounded set $B$ in $F$ there is $varepsilon>0$ with $varepsilon Bsubseteq U-x$ .

The Frölicher−Kriegl theory is essentially a bornological theory. One may observe that in Frölicher's and Kriegl's book one uses a canonical topology corresponding to the bornology, namely the strongest, bornological one, whereas in Kriegl's and Michor's book one allows any locally convex topology with the same bounded sets. In this sense, the KM−approach to smoothness is a bit floppy since the spaces are topological but bornology is the only one that matters.