Open Mapping Principle Research Articles

On the pillars of Functional Analysis

Many authors consider that the main pillars of Functional Analysis are the Hahn–Banach Theorem, the Uniform Boundedness Principle and the Open Mapping Principle. The first one is derived from Zorn’s Lemma, while the latter two usually are obtained from Baire’s Category Theorem. In this paper we show that these three pillars should be either just two or at least eight, since the Uniform Boundedness Principle, the Open Mapping Principle and another five theorems are equivalent, as we show in a very elemental way. Since one can give an almost trivial proof of the Uniform Boundedness Principle that does not require the Baire’s theorem, we conclude that this is also the case for the other equivalent theorems that, in this way, are simultaneously proved in a simple, brief and concise way that sheds light on their nature.

How Fundamental is the Fundamental Theorem of Algebra?

SummaryWe give several non-obvious proofs of the fundamental theorem of algebra, coming from different parts of mathematics. These include the idea of homotopy, Hurwitz’s theorem, Rouché’s theorem, the open mapping principle, and the argument principle.

The open mapping principle for partial actions of Polish groups

We present a extension of the classical open mapping principle and Effros' theorem for Polish group actions to the context of partial group actions.

Lyusternik--Graves Theorems for the Sum of a Lipschitz Function and a Set-valued Mapping

In a paper of 1950 Graves proved that for a function $f$ acting between Banach spaces and an interior point $\bar x$ in its domain, if there exists a continuous linear mapping $A$ which is surjective and the Lipschitz modulus of the difference $f-A$ at $\bar x$ is sufficiently small, then $f$ is (linearly) open at $\bar x$. This is an extension of the Banach open mapping principle from continuous linear mappings to Lipschitz functions. A closely related result was obtained earlier by Lyusternik for smooth functions. In this paper, we obtain Lyusternik--Graves theorems for mappings of the form $f+F$, where $f$ is a Lipschitz continuous function around $\bar x$ and $F$ is a set-valued mapping. Roughly, we give conditions under which the mapping $f+F$ is linearly open at $\bar x$ for $\bar y$ provided that for each element $A$ of a certain set of continuous linear operators the mapping $f(\bar{x}) +A(\cdot - \bar{x}) + F$ is linearly open at $\bar x$ for $\bar y$. In the case when $F$ is the zero mapping, as corollaries we obtain the theorem of Graves as well as open mapping theorems by Pourciau and Páles, and a constrained open mapping theorem by Cibulka and Fabian. From the general result we also obtain a nonsmooth inverse function theorem proved recently by Cibulka and Dontchev. Application to Nemytskii operators and a feasibility mapping in control are presented.

Effros, Baire, Steinhaus and non-separability

We give a short proof of an improved version of the Effros Open Mapping Principle via a shift-compactness theorem (also with a short proof), involving ‘sequential analysis’ rather than separability, deducing it from the Baire property in a general Baire-space setting (rather than under topological completeness). It is applicable to absolutely-analytic normed groups (which include complete metrizable topological groups), and via a Steinhaus-type Sum-set Theorem (also a consequence of the shift-compactness theorem) includes the classical Open Mapping Theorem (separable or otherwise).

Group action and shift-compactness

Shift-compactness has recently been found to be the foundation stone of classical, as well as topological, regular variation; most recently it has come again to prominence in new proofs of the Effros Open Mapping Principle of group action, another ingredient of topological regular variation. Using the real line under the Euclidean and density topologies as a paradigm, we develop group-action versions of shift-compactness theorems for Baire groups acting on Baire spaces under metrizable topologies and under certain refinements of these. One aim is to pursue constructive approaches rather than rely on plain Baire-category methods (so keeping more to the Banach–Mazur strategic approach). Along the way we uncover three new coarse topologies for groups of homeomorphisms. A second purpose is to establish limitations of the shift-compactness methodology.

On the Bartle-Graves theorem

The Bartle-Graves theorem extends the Banach open mapping principle to a family of linear and bounded mappings, thus showing that surjectivity of each member of the family is equivalent to the openness of the whole family. In this paper we place this theorem in the perspective of recent concepts and results, and present a general Bartle-Graves theorem for set-valued mappings. As application, we obtain versions of this theorem for mappings defined by systems of inequalities, and for monotone variational inequalities.

A strong uniform boundedness principle in Banach spaces

We discuss the term thick set. With the help of this term we deduce a strong Uniform Boundedness Principle valid for all Banach spaces. As an application we give a new proof of Seever's theorem. 1. DEFINITIONS AND PRELIMINARY RESULTS In this paper X is a Banach space. In [3] the notion of a set was introduced. It turns out that if a bounded set C X is then for any Banach space Y, any T E L(Y, X) has to be Further, not is the weakest general condition such that onto A implies onto X. It is natural to ask for the weakest general condition to be put on a bounded set to assure that pointwise boundedness of a family of operators in L(X, Y) implies uniform boundedness. The Banach-Steinhaus theorem tells us that second category is a sufficient condition, but the Nikodrm boundedness theorem shows that a uniform boundedness is true under weaker conditions, in particular spaces at least. Our main result in this paper is that not is the condition sought for this problem as well. Since, for our purposes at least, not seems to be more important than thin, we prefer to use the terms and thick instead of and not thin. Definition 1.1. bounded set B C X is called non-norming for X* if inf sup If(x) I 0. fCSx* xCzB bounded set C C X is called thin if it is the countable union of an increasing family of sets which are non-norming for X*. If a bounded set C is we will call it thick. In [3] Kadets and Fonf showed that there is an open mapping principle associated to thickness: Theorem 1.1. Suppose B C X is bounded. Then the following are equivalent statements: 1) For all Banach spaces Y and all T E L2(Y, X) with TY D B, we have TY = Received by the editors October 9, 1997 and, in revised form, May 13, 1998 and June 1, 1999. 2000 Mathematics Subject Classification. Primary 46B20; Secondary 28A33.

Subtraction Theorems and Approximate Openness for Multifunctions: Topological and Infinitesimal Viewpoints

An approximate open mapping principle for multifunctions between metric spaces with closed graphs is given. It is shown that infinitesimal assumptions involving tangent cones and approximate tangent cones ensure that openness results can be derived from the general principle. Several examples and counter-examples enlight the validity and the interplay of the results. Applications are given to the lower semicontinuity of the intersection of two multifunctions.

Normed linear relations: domain decomposability, adjoint subspaces, and selections

This paper is a study of normed linear relations with emphasis on results and properties that do not necessarily hold, or have no counterparts, for arbitrary normed convex processes. Sufficient conditions are given under which the norm of a linear relation M is equal to the norm of some operator part (i.e., a single-valued linear selection) of M. Generalizations of the closed-graph theorem and the open-mapping principle for linear relations are established in terms of linear selections. The norm of the adjoint subspace of a closed linear relation M is shown to be equal to the norm of M without any restrictions on the domain of M. The rich structure of normed linear relations is uncovered via four fundamental concepts and tools: algebraic, topological, and proximinal operator parts, and the adjoint subspace of a linear relation. Notions of algebraic, topological, and orthogonal domain decomposability of linear relations in Banach spaces are studied, and the relationships between these notions of domain decomposability and the corresponding notions of operator parts are determined.

Some inverse mapping theorems

We prove several first and high order inverse mapping theorems for set-valued maps from a complete metric space to a Banach space and study the stability of the open mapping principle. The obtained results allow to investigate questions of controllability of finite and infinite dimensional control systems, necessary conditions for optimality, implicit function theorem, stability of constraints with respect to a parameter. Applications to problems of optimization, control theory and nonsmooth analysis are provided.

Verifiable necessary and sufficient conditions for openness and regularity of set-valued and single-valued maps

We provide several equivalences to the regularity of closed set-valued maps around a point in general metric settings. In particular, an easy to verify approximate openness notion is shown equivalent to regularity. A simple specialization of our theorem strengthens Frankowska's novel “open mapping principle.”

An open mapping principle for set-valued maps

We prove an open mapping principle for set-valued maps on Banach spaces using “ kth order variations”.

An Elementary Counterexample to the Open Mapping Principle for Bilinear Maps

In [2], Rudin asked whether a continuous bilinear map from the product of two Banach spaces onto a Banach space must be open at the origin; i.e., whether under such a map the image of every neighborhood of zero must contain a neighborhood of zero. Recently, Cohen [1] showed that the answer to the general question was in the negative. However, his counterexample was somewhat involved and left the issue unresolved for bilinear maps on Hilbert spaces. The purpose of this note is to show that the open mapping principle for bilinear maps, as described above, fails even in the finite dimensional case.

An elementary counterexample to the open mapping principle for bilinear maps

In [2], Rudin asked whether a continuous bilinear map from the product of two Banach spaces onto a Banach space must be open at the origin; i.e., whether under such a map the image of every neighborhood of zero must contain a neighborhood of zero. Recently, Cohen [1] showed that the answer to the general question was in the negative. However, his counterexample was somewhat involved and left the issue unresolved for bilinear maps on Hilbert spaces. The purpose of this note is to show that the open mapping principle for bilinear maps, as described above, fails even in the finite dimensional case.