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• "Lipschitz continuity"
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In this article, using the Mizar system , , we discuss the continuity of bounded linear operators on normed linear spaces. In the first section, it is discussed that bounded linear operators on normed linear spaces are uniformly continuous and Lipschitz continuous. Especially, a bounded linear operator on the dense subset of a complete normed linear space has a unique natural extension over the whole space. In the next section, several basic currying properties are formalized.

In the last section, we formalized that continuity of bilinear operator is equivalent to both Lipschitz continuity and local continuity. We referred to , , and  in this formalization.

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In this article, we formalize in Mizar ,  the existence and uniqueness part of the implicit function theorem. In the first section, some composition properties of Lipschitz continuous linear function are discussed. In the second section, a definition of closed ball and theorems of several properties of open and closed sets in Banach space are described. In the last section, we formalized the existence and uniqueness of continuous implicit function in Banach space using Banach fixed point theorem. We referred to , , and  in this formalization.

## Summary

In this article, various definitions of contuity of multilinear operators on normed linear spaces are discussed in the Mizar formalism ,  and . In the first chapter, several basic theorems are prepared to handle the norm of the multilinear operator, and then it is formalized that the linear space of bounded multilinear operators is a complete Banach space.

In the last chapter, the continuity of the multilinear operator on finite normed spaces is addressed. Especially, it is formalized that the continuity at the origin can be extended to the continuity at every point in its whole domain. We referred to , , ,  in this formalization.

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The main aim of this article is proving properties of bilinear operators on normed linear spaces formalized by means of Mizar . In the first two chapters, algebraic structures  of bilinear operators on linear spaces are discussed. Especially, the space of bounded bilinear operators on normed linear spaces is developed here. In the third chapter, it is remarked that the algebraic structure of bounded bilinear operators to a certain Banach space also constitutes a Banach space.

In the last chapter, the correspondence between the space of bilinear operators and the space of composition of linear opearators is shown. We referred to , , ,  and  in this formalization.

## Summary

In the first chapter, the notion of multilinear operator on real linear spaces is discussed. The algebraic structure  of multilinear operators is introduced here. In the second chapter, the results of the first chapter are extended to the case of the normed spaces. This chapter shows that bounded multilinear operators on normed linear spaces constitute the algebraic structure. We referred to , , ,  in this formalization.

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In this article, we formalize differentiability of implicit function theorem in the Mizar system , . In the first half section, properties of Lipschitz continuous linear operators are discussed. Some norm properties of a direct sum decomposition of Lipschitz continuous linear operator are mentioned here.

In the last half section, differentiability of implicit function in implicit function theorem is formalized. The existence and uniqueness of implicit function in  is cited. We referred to , , and  in the formalization.

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In this article, we formalize topological properties of real normed spaces. In the first part, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed subspace. Then we discuss linear functions between real normed speces. Several kinds of subspaces induced by linear functions such as kernel, image and inverse image are considered here. The fact that Lipschitz continuity operators preserve convergence of sequences is also refered here. Then we argue the condition when real normed subspaces become Banach’s spaces. We also formalize quotient vector space. In the last session, we argue the properties of the closure of real normed space. These formalizations are based on (p.3-41),  and (p.3-67).

## Abstract

The paper offers a new approach to handling difficult parametric inverse problems in elasticity and thermo-elasticity, formulated as global optimization ones. The proposed strategy is composed of two phases. In the first, global phase, the stochastic hp-HGS algorithm recognizes the basins of attraction of various objective minima. In the second phase, the local objective minimizers are closer approached by steepest descent processes executed singly in each basin of attraction. The proposed complex strategy is especially dedicated to ill-posed problems with multimodal objective functionals. The strategy offers comparatively low computational and memory costs resulting from a double-adaptive technique in both forward and inverse problem domains. We provide a result on the Lipschitz continuity of the objective functional composed of the elastic energy and the boundary displacement misfits with respect to the unknown constitutive parameters. It allows common scaling of the accuracy of solving forward and inverse problems, which is the core of the introduced double-adaptive technique. The capability of the proposed method of finding multiple solutions is illustrated by a computational example which consists in restoring all feasible Young modulus distributions minimizing an objective functional in a 3D domain of a photo polymer template obtained during step and flash imprint lithography.

-791 Soravia, P. - Pursuit-evasion problems and viscosity solutions of Isaacs equations , SIAM J. Control Optim., 31 (1993), 604-623. Veliov, V.M. - Lipschitz continuity of the value function in optimal control , J. Optim. Theory Appl., 94 (1997), 335-363. Wolenski, P.; Zhuang, Y. - Proximal analysis and the minimal time function , SIAM J. Control Optim., 36 (1998), 1048-1072.

strong solution . Note 10 The existence of solutions of the above Protter’s type stochastic inclusion ensues from Lipschitz continuity of order-convex selections f of F, obtained in Theorem 10. But we know much more on the regularity of such selections by the constructive method of the proof of Theorem 10. They can be taken as convex envelopes (second generalized Fenchel transforms) of a function V ¯ F ( t , x ) = Π F ( t , x ) ( V ( t , x ) ) $\bar{V}_F(t,x)=\Pi _{F(t,x)}(V(t,x))$ . It means that selections f ( t , ·) are “greatest” order-convex selections of F  