Functional Analysis Homework Solutions: Tips and Tricks from Conway's Book
As a helpful review, here are a variety of problem and solutions to exercises from an introductory functional analysis class. This is the second installment in a series of functional analysis exercises (Part 1, Part 2, Part 3, Part 4)
Conway Functional Analysis Homework Solutions
@articleBredies2013,abstract = The ill-posed problem of solving linear equations in the space of vector-valued finite Radon measures with Hilbert space data is considered. Approximate solutions are obtained by minimizing the Tikhonov functional with a total variation penalty. The well-posedness of this regularization method and further regularization properties are mentioned. Furthermore, a flexible numerical minimization algorithm is proposed which converges subsequentially in the weak* sense and with rate 𝒪(n-1) in terms of the functional values. Finally, numerical results for sparse deconvolution demonstrate the applicability for a finite-dimensional discrete data space and infinite-dimensional solution space.,author = Bredies, Kristian, Pikkarainen, Hanna Katriina,journal = ESAIM: Control, Optimisation and Calculus of Variations,keywords = inverse problems; vector-valued finite Radon measures; Tikhonov regularization; delta-peak solutions; generalized conditional gradient method; iterative soft-thresholding; sparse deconvolution; ill-posed problem; linear equation; Hilbert space; numerical results,language = eng,number = 1,pages = 190-218,publisher = EDP-Sciences,title = Inverse problems in spaces of measures,url = ,volume = 19,year = 2013,
TY - JOURAU - Bredies, KristianAU - Pikkarainen, Hanna KatriinaTI - Inverse problems in spaces of measuresJO - ESAIM: Control, Optimisation and Calculus of VariationsPY - 2013PB - EDP-SciencesVL - 19IS - 1SP - 190EP - 218AB - The ill-posed problem of solving linear equations in the space of vector-valued finite Radon measures with Hilbert space data is considered. Approximate solutions are obtained by minimizing the Tikhonov functional with a total variation penalty. The well-posedness of this regularization method and further regularization properties are mentioned. Furthermore, a flexible numerical minimization algorithm is proposed which converges subsequentially in the weak* sense and with rate 𝒪(n-1) in terms of the functional values. Finally, numerical results for sparse deconvolution demonstrate the applicability for a finite-dimensional discrete data space and infinite-dimensional solution space.LA - engKW - inverse problems; vector-valued finite Radon measures; Tikhonov regularization; delta-peak solutions; generalized conditional gradient method; iterative soft-thresholding; sparse deconvolution; ill-posed problem; linear equation; Hilbert space; numerical resultsUR - ER -
This course will build the basic apparatus of measure theory and integration. Topics will include abstract measure and integration, convergence theorems for integration, Fubini's theorem, the Radon-Nikodym theorem, the Riesz-Markov representation theorem, and duals of certain function spaces. Time permitting we will do some functional analysis, with a look at software from Hilbert spaces, Banach spaces, and topological vector spaces, and hardware from Sobolev inequalities.