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... As a basic principle of functional analysis, the closed graph theorem has obtained a series of important improvements [1][2][3] since its first version was established by Banach [4] . However, all these results considered only the linear mappings. ...

In this paper, a new nonlinear closed graph theorem is established, whose result improves substantially a recent-known important
conclusion.

We obtain a new closed graph theorem which is a substantial improvement of a recent result.

This chapter focuses on the Slowikowski, Raikov and De Wilde closed graph theorems. The vector spaces used in the chapter, are defined over the field Ղ of real or complex numbers. The term, “space” means separated topological vector space, unless the contrary is specifically stated. If Ω is a non-empty open subset of the n-dimensional euclidean space, then the Schwartz space ҟ′(Ω) endowed with the strong topology belongs to this class. The chapter also studies the classes of spaces related with this conjecture. The class of Słowikowski spaces contains the F-spaces and it is stable with respect to the operations that include: countable topological direct sums, closed subspaces, countable topological products, and continuous linear images.

Let AT be a convergence space and C(X) the R-algebra of all continuous real-valued functions on X, equipped with the continuous convergence structure. If the natural map from X into C(C(X)) is an embedding, then X is said to be a c-space. With each space X there is associated the c-modification cX which is a c-space with the property C(X) = C(cX), This leads to the following theorems which are valid for any convergence space X: (1) C(X) is a topological space iff cX is locally compact; (2) C(X) is locally compact iff cX is finite.

An introduction to locally convex inductive limits, in: Functional Analysis and its Applications

- K.-D Bierstedt

K.-D. Bierstedt, An introduction to locally convex inductive limits, in: Functional Analysis and its Applications, Nice, 1986, ICPAM Lecture Notes, World Scientific, Singapore, 1988.

- A Grothendieck

A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math.
Soc. 16 (1955).