Notes on the Banach-Necas-Babuska theorem and Kato's minimum modulus of operators
This is a pedagogical review for specialists in numerical analysis, providing a clear exposition of known theorems; it is incremental.
This note reviews proofs of the Closed Range Theorem and Banach-Necas-Babuska (BNB) theorem using Kato's minimum modulus, and applies the BNB theorem to prove well-posedness of parabolic equations. No new results are presented.
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.