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Let F be an endofunctor of a category C. We prove isomorphism theorems for F -coalgebras under condition that the underlying category C is exact; that is, regular with exact sequences. Also, F is not assumed to preserve pullbacks.

Let A be a locally $m$-convex Fréchet algebra. We give a necessary and sufficient condition for a cyclic Fréchet $A-$module $X=A+/I$ to be strictly flat, generalizing thereby a criterion of Helemskii and Sheinberg. To this end, we introduce a notion of "locally bounded approximate identity" (a locally b.a.i. for short), and we show that $X$ is strictly flat if and only if the ideal I has a right locally b.a.i. Next we apply this result to amenable algebras and show that a locally $m$-convex Fréchet algebra $A$ is amenable if and only if $A$ is isomorphic to a reduced inverse limit of amenable Banach algebras. We also extend a number of characterizations of amenability obtained by Johnson and by Helemskii and Sheinberg to the setting of locally $m$-convex Fréchet algebras. As a corollary, we show that Connes and Haagerup's theorem on amenable $C*$-algebras and Sheinberg's theorem on amenable uniform algebras hold in the Fréchet algebra case. We also show that a quasinormable locally $m$-convex Fréchet algebra has a locally b.a.i. if and only if it has a b.a.i. On the other hand, we give an example of a commutative, locally $m$-convex Fréchet-Montel algebra which has a locally b.a.i., but does not have a b.a.i.

A. Grothendieck proved at the end of his thesis that the space $\mathcal{O}_M$ of slowly increasing functions and the space $\mathcal{O}_{C}'$ of rapidly decreasing distributions are bornological. Grothendieck's proof relies on the isomorphy of these spaces to a sequence space and we present the first proof that does not utilize this fact by using homological methods and, in particular, the derived projective limit functor.

A shorter proof is given for a theorem of Domanski and Mastylo characterizing the pairs $(E,F)$ of Frechet-Hilbert spaces with the property that every exact sequence $0\to F\to G\to E\to 0$ of Frechet-Hilbert spaces splits. The results on acyclicity of inductive spectra of metrizable locally convex spaces which we use are also presented with proofs.

We consider the problem of real analytic parameter dependence of solutions of the linear partial differential equation $P(D)u=f$, i.e., the question if for every family $(f\sb\lambda)\subseteq \mathscr{D}'(\Omega)$ of distributions depending in a real analytic way on $\lambda\in U$, $U$ a real analytic manifold, there is a family of solutions $(u\sb\lambda)\subseteq \dio$ also depending analytically on $\lambda$ such that $$ P(D)u\sb\lambda=f\sb\lambda\qquad \text{for every $\lambda\in U$}, $$ where $\Omega\subseteq \mathbb{R}\sp d$ an open set. For general surjective variable coefficients operators or operators acting on currents over a smooth manifold we give a solution in terms of an abstract ``Hadamard three circle property'' for the kernel of the operator. The obtained condition is evaluated giving the full solution (usually in terms of the symbol) for operators with constant coefficients and open (convex) $\Omega\subseteq\mathbb{R}\sp d$ if $P(D)$ is of one of the following types: 1) elliptic, 2) hypoelliptic, 3) homogeneous, 4) of two variables, 5) of order two or 6) if $P(D)$ is the system of Cauchy-Riemann equations. An analogous problem is solved for convolution operators of one variable. In all enumerated cases, it follows that the solution is in the affirmative if and only if $P(D)$ has a linear continuous right inverse which shows a striking difference comparing with analogous smooth or holomorphic parameter dependence problems. The paper contains the whole theory working also for operators on Beurling ultradistributions $\mathscr{D}'\sb{(\omega)}$. We prove a characterization of surjectivity of tensor products of general surjective linear operators on a wide class of spaces containing most of the natural spaces of classical analysis.