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.

翻译：这份说明是为京都大学的演讲编写的(RIMS讲习班:“数字分析中的艺术状况:理论、方法和应用”,2017年11月8日至10日,2017年11月8日至10日),该讲座介绍了根据作者早些时候的一份文件(arXiv:1710.10543)对抛物线方程式不连续加列金时间步法的变式分析(arXiv:1710.10543);我还介绍了Banach-Necas-Babuska(BNB) Theorem或Babuska-Lax-Milgram(BLM)的理论,作为我们分析的关键理论。为证明,BNB thetheorem的理论基础并非原始研究论文。T. Kato提出的操作者最低模范方法的证明是有用的。本说明回顾了根据Kato的想法对封闭区域理论和BNBTheorem的证据。此外,我还介绍了BNB theorem对抛物方方程式的应用。BNBthe the presseded in the private real produment is fird real shal deplain report.