  This note was prepared for a lecture given at Kyoto University (RIMS
Workshop: "The State of the Art in Numerical Analysis: Theory, Methods, and
Applications", November 8-10, 2017). That lecture described the variational
analysis of the discontinuous Galerkin time-stepping method for parabolic
equations based on an earlier paper by the author (arXiv:1710.10543). I also
presented the Banach-Necas-Babuska (BNB) Theorem or the Babuska-Lax-Milgram
(BLM) Theorem as the key theorem of our analysis. For proof of the BNB theorem,
it is useful to introduce the minimum modulus of operators by T. Kato. This
note presents a review of the proofs of Closed Range Theorem and BNB Theorem
following the idea of Kato. Moreover, I present an application to BNB theorem
to parabolic equations. The well-posedness is proved by BNB theorem. This note
is not an original research paper. It includes no new results. This is a
revised manuscript and several incorrect descriptions in the original version
are fixed.
