Is surjectivity an open condition for linear maps on Frechet spaces? Given a linear map $L:V\to W$ between finite dimensional vector spaces it is well-known that if $L$ is surjective then any sufficiently close by (with respect to some norm) by linear map remains surjective.
For Banach spaces invertability of a linear map still implies invertability of linear maps close by, so I suspect that close by linear maps to a surjective one still remain surjective.
In Frechet spaces this openess no longer holds true for invertability. However, the examples for the failure which I have seen so far all fail to be injective. So I am looking for an example where surjectivity fails, i.e. a continuous family of linear maps $L:P\times F\to G$ between Frechet spaces $P,F,G$ ($L(p,\cdot):F\to G$ is linear for each $p\in P$) such that $L(0,\cdot):F\to G$ is surjective but there is a sequence $(p_n)\to 0\in P$ such that $L(p_n)$ is not surjective. Alternatively, a proof that surjectivity is indeed an open condition, if that would be true.
 A: The set of surjective operators between Frechet spaces does not need to be open in general (I assume that the topology on the space of continuous operators is the topology of uniform convergence on bounded sets - the strongest among standard topologies on this space).
The following example comes from the theory of differential operators with constant coefficients, where a lot of results on surjectivity are known. Let $\Omega = \{(x,y) \in \mathbb R^2: |x|,|y| < 1\} \setminus \{(x,0): x \le 0\} \subset \mathbb R^2$ be the rectangle cut by an interval. Let $L_h$ denote the directional derivative along $h = (h_1, h_2) \in \mathbb R^2$, i.e. $L_h f = h_1 \partial f / \partial x + h_2 \partial f / \partial y$. It is clear that $L_h$ is a continuous operator on the Frechet space $C^\infty(\Omega)$. Moreover, it can be proved that $L_h$ depends continuously on $h$ and is surjective if and only if $h_1 \ne 0$ and $h_2 = 0$. Thus, the set $\{h \in \mathbb R^2: L_h\;\text{is surjective}\}$ is not open, proving that the set of surjective operators on $C^\infty(\Omega)$ is not open.
In order to prove the surjectivity condition for $L_h$ see Theorem 10.8.3 from Analysis of linear partial differential operators II by Hormander.
