Linear Equations in Banach Spaces

1982 ◽  
Author(s):  
S. G. Krein
Keyword(s):  
Banach Spaces ◽  
Linear Equations

scholarly journals On Leżański's determinants of linear equations in Banach spaces

1953 ◽  
Vol 14 (1) ◽  
pp. 24-48 ◽  
Author(s):  
R. Sikorski
Keyword(s):  
Banach Spaces ◽  
Linear Equations

scholarly journals A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations

2018 ◽  
Vol 26 (5) ◽  
pp. 639-646 ◽  
Author(s):  
Jens Flemming
Keyword(s):  
Banach Spaces ◽  
Convergence Rates ◽  
A Priori ◽  
Linear Equations ◽  
Rate Function ◽  
Bregman Distances ◽  
Error Measures ◽  
Ill Posed ◽  
General Convex ◽  
Linear Operator Equations

Abstract We consider Tikhonov-type variational regularization of ill-posed linear operator equations in Banach spaces with general convex penalty functionals. Upper bounds for certain error measures expressing the distance between exact and regularized solutions, especially for Bregman distances, can be obtained from variational source conditions. We prove that such bounds are optimal in case of twisted Bregman distances, a specific a priori parameter choice, and low regularity of the exact solution, that is, the rate function is also an asymptotic lower bound for the error measure. This result extends existing converse results from Hilbert space settings to Banach spaces without adhering to spectral theory.

scholarly journals The Fredholm Theory of Linear Equations in Banach spaces

1953 ◽  
Vol 13 (2) ◽  
pp. 244-276 ◽  
Author(s):  
T. Leżański
Keyword(s):  
Banach Spaces ◽  
Linear Equations ◽  
Fredholm Theory
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