Exactness of completed tensor product of nuclear spaces

Unfortunately, this is not true in general.

Let $E$, $F$, $G$ and $H$ be complete nuclear lcs such that $$ 0 longrightarrow E stackrel{S}{longrightarrow} F stackrel{T}{longrightarrow} G longrightarrow 0 $$ is a strict short exact sequence. Note that the sequence $$ 0 longrightarrow E mathbin{hatotimes} H stackrel{S mathbin{hatotimes} text{id}_H}{longrightarrow} F mathbin{hatotimes} H stackrel{T mathbin{hatotimes} text{id}_H}{longrightarrow} G mathbin{hatotimes} H longrightarrow 0 $$ is a strict short exact sequence if and only if $T mathbin{hatotimes} text{id}_H : F mathbin{hatotimes} H to G mathbin{hatotimes} H$ is surjective; everything else is automatic by the mapping properties of the injective resp. projective topology.

Let $w$ and $w*$ denote the weak and weak-$*$ topologies, and let $T' : G_{w*}' to F_{w*}'$ denote the adjoint of $T$. It follows from [Sch99, §IV.9.4] that $F mathbin{hatotimes} H cong mathfrak{B}_e(F_{w*} times H_{w*})$. The latter is linearly isomorphic with $mathfrak{L}(F_{w*}' , H_w)$, and the following diagram commutes: $$require{AMScd} begin{CD} F mathbin{hatotimes} H @>T mathbin{hatotimes} text{id}_H >> G mathbin{hatotimes} H \ @VV V @VV V \ mathfrak{L}(F_{w*}' , H_w) @> R mapsto RT' >> mathfrak{L}(G_{w*}' , H_w) \ end{CD}$$ Since $G cong F/E$, we have $G' = E^perp$, and the weak-$*$ topology on $G'$ coincides with the relative $F_{w*}'$ topology of $E^perp$. Thus, we see that $T mathbin{hatotimes} text{id}_H$ is surjective if and only if every weak-$*$-to-weak continuous operator $E^perp to H$ can be extended to $F'$. This is not always the case:

Counterexample. Let $E$ be a closed, non-complemented subspace of a nuclear Fréchet space $F$.¹ Furthermore, let $G := F/E$, and let $H := G_beta'$ be the strong dual of $G$. Then $E$, $F$ and $G$ are nuclear Fréchet spaces (so in particular reflexive), and $H$ is a complete, nuclear (DF)-space.

In a Fréchet space, all weakly complemented subspaces are complemented, so $E$ is not weakly complemented in $F$. By duality, $E^perp = G'$ is not weak-$*$ complemented in $F'$. Since $G$ is reflexive, the weak and weak-$*$ topologies on $H = G_beta'$ coincide, so we have $text{id}_H in mathfrak{L}(G_{w*}' , H_w)$. Since $G'$ is not weak-$*$ complemented, the map $text{id}_H in mathfrak{L}(G_{w*}' , H_w)$ has no extension in $mathfrak{L}(F_{w*}' , H_w)$.

Convesely, if $E$ is (weakly) complemented, then $T mathbin{hatotimes} text{id}_H$ is always surjective.

_{¹: Concrete examples of this kind are given in this answer, or in [MV97, Exercise 31.4], or in [DM76], among others.}

References.

[DM76] Plamen Djakov, Boris Mitiagin, Modified construction of nuclear Fréchet spaces without basis, Journal of Functional Analysis, vol. 23 (1976), issue 4, pp. 415–433. DOI: 10.1016/0022-1236(76)90066-5.

[MV97] Reinhold Meise, Dietmar Vogt, Introduction to Functional Analysis (1997), Oxford Graduate Texts in Mathematics, Clarendon Press, Oxford.

[Pie72] Albrecht Pietsch, Nuclear Locally Convex Spaces (1972), Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, Berlin.

[Sch99] H.H. Schaefer, M.P. Wolff (translator), Topological Vector Spaces, Second Edition (1999), Springer Graduate Texts in Mathematics, Springer, New York.