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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2022 • Question 6

Question

Math 6210 • Real Analysis Fall 2022 • Question 6

State the open mapping theorem for Banach spaces and use it to prove the following: Let \((B, \|\cdot \|_{B})\) and \((C, \|\cdot \|_{C})\) be Banach spaces and \(A : B \to C\) be continuous and linear. If \(A\) is a bijection then \(A^{-1}\) is continuous.

Raw LaTeX

State the open mapping theorem for Banach spaces and use it to prove
the following: Let \((B, \|\cdot\|_{B})\) and \((C, \|\cdot\|_{C})\) be Banach spaces and \(A : B \to C\)
be continuous and linear. If \(A\) is a bijection then \(A^{-1}\) is continuous.

Topic Tags

Banach Spaces Open Mapping and Closed Graph Theorems