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- Let X, Y be Banach spaces. We call an operator TE B(X,Y) invertible if there exists an operator T-l e B(Y, X) such that T-'T = Ix and TT-l = Iy. Show that the following statements are equivalent: (a) T is invertible. (b) T is bijective. (c) T has dense range and T is bounded below (i.e., there exists c > 0 such that |T(x)| > c[x| for all r e X).

- Let X, Y be Banach spaces. We call an operator TE B(X,Y) invertible if there exists an operator T-l e B(Y, X) such that T-'T = Ix and TT-l = Iy. Show that the following statements are equivalent: (a) T is invertible. (b) T is bijective. (c) T has dense range and T is bounded below (i.e., there exists c > 0 such that |T(x)| > c[x| for all r e X).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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- Let X, Y be Banach spaces. We call an operator TE B(X,Y) invertible if there exists an operator T-' e B(Y, X) such that T-IT = Ix and TT-1 = Iy. Show that the following statements are equivalent: (a) T is invertible. (b) T is bijective. (c) T has dense range and T is bounded below (i.e., there exists c > 0 such that |T(x)| > c||r| for all x e X).
Transcribed Image Text:- Let X, Y be Banach spaces. We call an operator TE B(X,Y) invertible if there exists an operator T-' e B(Y, X) such that T-IT = Ix and TT-1 = Iy. Show that the following statements are equivalent: (a) T is invertible. (b) T is bijective. (c) T has dense range and T is bounded below (i.e., there exists c > 0 such that |T(x)| > c||r| for all x e X).
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