Later on, we will however assume that all participants are familiar with measure and integration theory at a workable level. Measure and integration theory is not a formal prerequisite, an intuitive knowledge of it will be enough in the beginning of the course. Knowledge of compact operators or reflexivity (topics covered in some introductory courses) is not a prerequisite. You should be familiar with these notions and results at a workable level before you take this course, which is not suitable as a first acquaintance with functional analysis. Keywords to test yourself: Cauchy sequence, equivalence of norms, operator norm, dual space, adjoint operator, Hahn-Banach theorems, Baire category theorem, closed graph theorem, open mapping theorem, uniform boundedness principle, inner product and Cauchy-Schwarz inequality, orthogonal decomposition of a Hilbert space related to a closed subspace, orthonormal basis and Fourier coefficients, adjoint operator, orthogonal projection, selfadjoint/unitary/normal operators. Basic knowledge of Banach and Hilbert spaces and bounded linear operators as is provided by introductory courses, and hence also of general topology and metric spaces. Is maximal monotone and Browder’s theorem implies that Equation (4) has a unique solution (such an extension exists by virtue of Zorn’s lemma). Under hypotheses as above, Equation (4) has a unique solution The following theorem asserts the existence and uniqueness of generalized solution of (4). Under the above hypotheses, there exist the dual mappingsīeing strictly monotone, single-valued, homogeneous, hemi-continuous and such that Is also a hemi-continuous monotone operator from X into We now deal with the stable method of computing values of the operator A at Is open or everywhere dense in X, or if A is maximal monotone, then a generalized solutionĬoincides with the corresponding solution We note that, if A is hemi-continuous and If A is an arbitrary monotone operator, we follow and understand a solution of (1) to be an elementĪ generalized solution of Equation (1). If A is a maximal monotone (possibly multi-valued). In - a class of monotone operators was singled out and, as an approximate method, the operator-regularization method was used.Īs it is known, a solution of (1) is understood to be an element These problems are important objects of investigation in the theory unstable problems. We consider the following three problemsģ) To compute values of the operator A at (possibly multi-valued) and y is a given element in Is a hemi-continuous monotone operator from X into Let X be a real strictly convex reflexive Banach space with the dual The Stable Method of Computing Values of Hemi-Continuous Monotone Operators The approximate values of the operator A atģ. In a similar way as above, the everywhere defined inverse Because of the uniqueness of decomposition (7), x is uniquely determined by z, and so the everywhere defined inverse , we have the uniquely determined decomposition Is a closed densely defined linear operator thenĪre complementary orthogonal subspaces of the Hilbert space The following lemma will be used in the proof of Theorem 2.2. To further simplify the presentation, we introduce the functions To establish the convergence of (3), it will be convenient to reformulate (3) asĪre known to be bounded everywhere defined linear operators and The minimization problem (1) has a unique solution Is also a closed densely defined unbounded linear operator from X into Y with domainįirst, we define the regularization functional Is a closed densely defined unbounded linear operator from a Hilbert space X into a Hilbert space Y with domain The Stable Method of Computing Values of Closed Densely Defined Unbounded Linear Operators In this paper we shall be concerned with the construction of a stable method of computing values of the operator A for the perturbations (2).Ģ. Until now, this problem is still an open problem. We should approximate values of A when we are given the approximations We now assume that both the operator A and In the another case, where A is a monotone operator from a real strictly convex reflexive Banach space X into its dual Moreover, the order of convergence results for Morozov has studied a stable method for approximating the value In the case, where A is a closed densely defined unbounded linear operator from a Hilbert space X into a Hilbert space Y, V. , we can see that the values of the operator A may not even be defined on the elements Therefore, the problem of computing values of an operator in the considered case is unstable. , where X and Y are normed spaces and A is unbounded, that is, there exists a sequence of elements Indeed, let A be a linear operator from X into Y with domain The stable computation of values of unbounded operators is one of the most important problems in computational mathematics.
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