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36
Banach space of bounded linear operators
 FORMALIZED MATHEMATICS
, 2003
"... On this article, the basic properties of linear spaces which are defined by the set of all linear operators from one linear space to another are described. Especially, the Banach space is introduced. This is defined by the set of all bounded linear operators. ..."
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On this article, the basic properties of linear spaces which are defined by the set of all linear operators from one linear space to another are described. Especially, the Banach space is introduced. This is defined by the set of all bounded linear operators.
On the decompositions of intervals and simple closed curves
 Journal of Formalized Mathematics
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On the minimal distance between sets in Euclidean space
 Journal of Formalized Mathematics
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On the Simple Closed Curve Property of the Circle and the Fashoda Meet Theorem for It
, 2003
"... First, we prove the fact that the circle is the simple closed curve, which was defined as a curve homeomorphic to the square. For this proof, we introduce a mapping which is a homeomorphism from 2dimensional plane to itself. This mapping maps the square to the circle. Secondly, we prove the Fasho ..."
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Cited by 3 (1 self)
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First, we prove the fact that the circle is the simple closed curve, which was defined as a curve homeomorphic to the square. For this proof, we introduce a mapping which is a homeomorphism from 2dimensional plane to itself. This mapping maps the square to the circle. Secondly, we prove the Fashoda meet theorem for the circle using this homeomorphism.
Definition of Integrability for Partial Functions from R to R and Integrability for Continuous Functions
"... Summary. In this article, we defined the Riemann definite integral of partial function from R to R. Then we have proved the integrability for the continuous function and differentiable function. Moreover, we have proved an elementary theorem of calculus. ..."
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Summary. In this article, we defined the Riemann definite integral of partial function from R to R. Then we have proved the integrability for the continuous function and differentiable function. Moreover, we have proved an elementary theorem of calculus.
On the Hausdorff distance between compact subsets
 Journal of Formalized Mathematics
"... Summary. In [2] the pseudometric distmax min on compact subsets A and B of a topological space generated from arbitrary metric space is defined. Using this notion we define the Hausdorff distance (see e.g. [6]) of A and B as a maximum of the two pseudodistances: from A to B and from B to A. We jus ..."
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Summary. In [2] the pseudometric distmax min on compact subsets A and B of a topological space generated from arbitrary metric space is defined. Using this notion we define the Hausdorff distance (see e.g. [6]) of A and B as a maximum of the two pseudodistances: from A to B and from B to A. We justify its distance properties. At the end we define some special notions which enable to apply the Hausdorff distance operator “HausDist ” to the subsets of the Euclidean topological space E n T.
Some Lemmas for the Jordan Curve Theorem
, 2003
"... I present some miscellaneous simple facts that are still missing in the library. The only common feature is that, most of them, were needed as lemmas in the proof of the Jordan curve theorem. ..."
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I present some miscellaneous simple facts that are still missing in the library. The only common feature is that, most of them, were needed as lemmas in the proof of the Jordan curve theorem.
Cages  the external approximation of Jordan’s curve
 Formalized Mathematics
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Scalar Multiple of Riemann Definite Integral
"... Summary. The goal of this article is to prove a scalar multiplicity of Riemann definite integral. Therefore, we defined a scalar product to the subset of real space, and we proved some relating lemmas. At last, we proved a scalar multiplicity of Riemann definite integral. As a result, a linearity of ..."
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Summary. The goal of this article is to prove a scalar multiplicity of Riemann definite integral. Therefore, we defined a scalar product to the subset of real space, and we proved some relating lemmas. At last, we proved a scalar multiplicity of Riemann definite integral. As a result, a linearity of Riemann definite integral was proven by unifying the previous article [11]. MML Identifier:INTEGRA2.