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The classical open mapping theorem in functional analysis certainly holds in the Banach space setting, and this is where I first encountered it. Slightly more advanced textbooks (e.g. Rudin's Functional Analysis) occasionally state and prove more general versions, for instance for $F$-spaces (i.e. metrizable by a complete translation-invariant metric, but not necessarily locally convex). Again, these qualify as classical results. It seems that the Banach property can be relaxed to a much weaker completeness condition, as long as the space is metrizable.

Even in the absence of metric, we still have a notion of completeness in general topological vector spaces (every Cauchy filter/net converges). However, the proof of the open mapping theorem for $F$-spaces relies on the Baire category theorem, so this really uses the complete metric in a non-trivial way. This leads me to the following question:

Question 1. Does the open mapping theorem generalize to arbitrary (not necessarily metrizable) complete topological vector spaces?

As the open mapping theorem comes in various forms (see e.g. Rudin Theorem 2.11), I may have to be a bit more specific. I am particularly interested in the following question:

Question 2. Are there complete topological vector spaces $X$ and $Y$ and a continuous linear surjection $T : X \to Y$ such that $T$ is not open?

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Question 1. Such results have been studied in detail—-a good reference is Köthe‘s monograph on topological linear spaces. You could also look up the concept of webbed spaces (de Wilde).

For question 2, you can take the space of bounded, continuous functions on the real line—-it has two distinct complete locally convex structures: that induced by the supremum norm and the strict topology which was introduced by R.C. Buck in the fifties.

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Another example for question 2 is the following: There are linear partial differential operators with constant coefficients (e.g., the wave operator) and open sets $\Omega \subseteq \mathbb R^3$ (e.g., the complement of a cone) such that $P(\partial)$ is surjective on the space $\mathscr E(\Omega)$ of smooth functions but not on the space $\mathscr D'(\Omega)$ of distributions. Then the range $X$ of the transposed $P(-\partial):\mathscr D(\Omega)\to \mathscr D(\Omega)$ is a closed subspace of the space $\mathscr D(\Omega)$ of test functions and it has two different complete locally convex topologies: The subspace topology from $\mathscr D(\Omega)$ and the strictly finer topology making $P(-\partial): \mathscr D(\Omega) \to X$ a topological isomorphism.

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A good reference for some of the most general versions of the open mapping theorem (and its cousin the closed graph theorem) is the book

Meise, Vogt: Introduction to Functional analysis, 1997

Among other things it contains the version with webbed spaces (based on deWildes work already mentioned)

To give another example of a continuous vector space isomorphism which is not open (whence not an isomorphism of locally convex spaces) let $J = [0,1]$ be a compact intervall and $I$ be some index set. For a locally convex vector space we consider $C(J,E)$ with the compact open topology (turning it into a locally convex space) and denote the direct sum of locally convex spaces by $\oplus_{i\in I}$ (i.e. we take the subset of the product consisting of all elements with only finitely many entries, with the locally convex direct sum topology) Then the mapping $$\Lambda \colon \bigoplus_{i\in I} C(J,\mathbb{R}) \rightarrow C(J,\bigoplus_{i\in I}\mathbb{R}), (f_i) \mapsto \sum_{i\in I} f_i$$ (where we identify every copy of $\mathbb{R}$ with a subspace of $\bigoplus_{i\in I}\mathbb{R}$ is easily seen to be a continuous vector space isomorphism. However, it now depends on the cardinality of $I$ whether or not $\Lambda$ is an isomorphism. If $I$ is countable, we obtain an isomorphism, for $I$ uncountable $\Lambda$ can not be open. The argument is a bit involved (and I learned it from S.A. Wegner and D. Vogt), whence I refer to C.3.6 Lemma and 5.4.12 Lemma of 1 where all the details are recorded.

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$\def\sp{\kern.3mm}\def\TT{{\mathscr T}}$Still further (counter)examples of the situation in Q2 are obtained from Propositions 4.4.3 (p. 81) and 6.6.7 (p. 111) and Example 6.10.L (p. 123) in Jarchow's Locally Convex Spaces as follows. Let $F_3=(X,\TT)$ be any infinite-dimensional Banach space, and let $F_r=(X,\TT_r)$ where $\TT_0$ is the finest linear topology for $X$. For $0<r\le 1$ let $\TT_r$ be the finest locally $r$−convex linear topology for $X$, and let $\TT_2$ be the "box topology" for $X$ obtained by identifying $X$ with the $B$−fold direct sum of the scalar field for any Hamel basis $B$ for $X$. Then taking any $r,s\in[\sp 0\sp,1\sp]\cup\{\sp 2\sp,3\sp\}$ with $r<s$ and $(r\sp,s)\not=(2\sp,3)$ the identity is a continuous linear map $F_r\to F_s$ of complete topological vector spaces that is not open. For $1\le r$ we even have locally convex spaces that, however, the OP did not require.

For Q1 I recommend to see Theorems 5.5.2 (p. 95) and 11.1.7 (p. 221) loc.cit.

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In fact, we have a partial opposite for general topological vector spaces:

(1) Theorem. Let $E$ be a vector space, and let $\mathcal T_1 \subseteq \mathcal T_2$ be linear topologies on $E$ such that $\mathcal T_2$ has a neighbourhood base at $0$ consisting of $\mathcal T_1$-closed sets. If $(E,\mathcal T_1)$ is complete, then so is $(E,\mathcal T_2)$.

For a proof, see [Köthe, §18,4.(4)], or [Jarchow, Theorem 3.3.4].

Using this, one can prove the following.

(2) Theorem. Let $(E,\mathcal T)$ be a Fréchet space, and let $(E',\mathcal T_\beta)$, $(E',\mathcal T_\mu)$ and $(E',\mathcal T_{pc})$ denote the (topological) dual of $E$ equipped with respectively the strong topology, the Mackey topology, and the topology of uniform convergence on the precompact subsets of $E$. Then one has $\mathcal T_{pc} \subseteq \mathcal T_\mu \subseteq \mathcal T_\beta$, and each of these locally convex topologies is complete.

For a proof, see [Köthe, §21,6.(4)], or [Jarchow, Proposition 9.5.3].

This has the following consequence:

(3) Example. Let $X$ be an infinite-dimensional Banach space. We claim that $X'$ admits two distinct (but comparable) complete locally convex topologies. We distinguish two cases.

  • If $X$ is reflexive, then its closed unit ball $B_X$ is absolutely convex and weakly compact, but not precompact in the original topology of $X$. Therefore the saturated classes defining $\mathcal T_{pc}$ and $\mathcal T_\mu$ are different, and consequently so are the topologies they define (cf. [Köthe, §21,1.(4)]). So we have $\mathcal T_{pc} \subsetneq \mathcal T_\mu$, with both of these being complete.

  • If $X$ is not reflexive, then $\mathcal T_\beta$ is not compatible with the dual pair $\langle X',X\rangle$, whereas $\mathcal T_{pc}$ and $\mathcal T_\mu$ are. So in this case we have $\mathcal T_\mu \subsetneq \mathcal T_\beta$, with both of these being complete.

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References.

[Köthe]: Gottfried Köthe, Topological Vector Spaces I, second revised printing (1983), Grundlehren der mathematischen Wissenschaften 159, Springer.

[Jarchow]: Hans Jarchow, Locally Convex Spaces (1981), Mathematische Leitfäden, Teubner.

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The study of open mapping theorems beyond Fréchet spaces seems to have been initiated by Vlastimil Pták in the 1950s. [Ptá53], [Ptá58] These papers are rather long and technical, but make for an interesting read (well, at least the English one).

At the very end of [Ptá58], some counterexamples are given for Question 2. Let $E = \ell^1(\mathbb{N}_1)$, and consider the following three linear topologies on $E$:

  • $\mathcal w$ is the locally convex topology given by the family of seminorms $\{p_\alpha\}_{\alpha \in A}$, where $A$ is the set of all sequences $\alpha = \{\alpha_k\}_{k=1}^\infty$ of real numbers satisfying $1 \geq \alpha_1 \geq \cdots \geq \alpha_k > 0$ (for all $k\in\mathbb{N}_1$) and $\lim_{k\to\infty} \alpha_k = 0$, and $p_\alpha(x) := \sum_{k=1}^\infty \alpha_k|x_k|$.
  • $\mathcal v$ is the usual norm topology on $\ell^1$.
  • $\mathcal u$ is the Mackey topology $\tau(E,E^*)$, where $E^*$ denotes the algebraic dual of $E$. (Equivalently, $\mathcal u$ is the finest locally convex topology on $E$; cf. [Sch99 , IV.3.3].)

Pták then shows that $\mathcal u \supset \mathcal v \supset \mathcal w$, where each of the inclusions is strict (in fact, each of these topologies yields a different topological dual space). He then proceeds to argue that each of these linear topologies is complete.

Pták uses this to further illustrate various notions and results introduced earlier in the same paper. (He provides notions of $B$-completeness and $B_r$-completeness, where $B$-complete $\implies$ $B_r$-complete $\implies$ complete. In the aforementioned example, he argues that $(E,\mathcal v)$ and $(E,\mathcal w)$ are not only complete but even $B$-complete, and $(E,\mathcal u)$ is an example of a space which is complete but not $B_r$-complete.)

I guess these are not the easiest examples for the problem at hand, but nevertheless a nice addition to the list.

References:

[Ptá53] Vlastimil Pták, О полных топологических линейных пространствах (On complete topological linear spaces), Czechoslovak Mathematical Journal, vol. 3 (1953), issue 4, pp. 301–364. https://dml.cz/dmlcz/100092 (In Russian.)

[Ptá58] Vlastimil Pták, Completeness and the open mapping theorem, Bulletin de la Société Mathématique de France, vol. 86 (1958), pp. 41–74. https://doi.org/10.24033/bsmf.1498

[Sch99], H.H. Schaefer (with M.P. Wolff), Topological Vector Spaces, Second Edition (1999), Graduate Texts in Mathematics 3, Springer.

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