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Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators by Randall Eubank, Tailen Hsing

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3.5 Operator inverses

A linear mapping c03-math-594 from a vector space c03-math-595 into a vector space c03-math-596 is one-to-one if c03-math-597 and onto if c03-math-598. When c03-math-599 is both one-to-one and onto it is said to be bijective. Bijective linear mappings are invertible. That is, there exists a linear mapping c03-math-600 from c03-math-601 to c03-math-602 such that c03-math-603 and c03-math-604 are the identity transformation on their respective spaces.

Suppose now that and are Banach spaces and that is bijective. ...

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