Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures and Tusi Mathematical Research Group)Khayyam Journal of Mathematics2423-47882120160101Closed Graph Theorems for Bornological Spaces811111752410.22034/kjm.2016.17524ENFedericoBambozziFakultät für Mathematik, Universität Regensburg, 93040 Regensburg, GermanyJournal Article20160930The aim of this paper is that of discussing closed graph theorems for bornological vector spaces in a self-contained way, hoping to make the subject more accessible to non-experts. We will see how to easily adapt classical arguments of functional analysis over $mathbb R$ and $mathbb C$ to deduce closed graph theorems for bornological vector spaces over any complete, non-trivially valued field, hence encompassing the non-Archimedean case too. We will end this survey by discussing some applications. In particular, we will prove De Wilde's Theorem for non-Archimedean locally convex spaces and then deduce some results about the automatic boundedness of algebra morphisms for a class of bornological algebras of interest in analytic geometry, both Archimedean (complex analytic geometry) and non-Archimedean.http://www.kjm-math.org/article_17524_5a0ac6149969bfb6b858ab7f5c5c3a47.pdf