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F. Riesz Theorem

100%

Narita K., Nakasho K., Shidama Y.

Formalized Mathematics

2017

tom 25

nr 3

179-184

In this article, we formalize in the Mizar system [1, 4] the F. Riesz theorem. In the first section, we defined Mizar functor ClstoCmp, compact topological spaces as closed interval subset of real numbers. Then using the former definition and referring to the article [10] and the article [5], we defined the normed spaces of continuous functions on closed interval subset of real numbers, and defined the normed spaces of bounded functions on closed interval subset of real numbers. We also proved some related properties. In Sec.2, we proved some lemmas for the proof of F. Riesz theorem. In Sec.3, we proved F. Riesz theorem, about the dual space of the space of continuous functions on closed interval subset of real numbers, finally. We applied Hahn-Banach theorem (36) in [7], to the proof of the last theorem. For the description of theorems of this section, we also referred to the article [8] and the article [6]. These formalizations are based on [2], [3], [9], and [11].

Spaces defined by the level function and their duals

100%

Sinnamon G.

Studia Mathematica

1994

tom 111

nr 1

19-52

The classical level function construction of Halperin and Lorentz is extended to Lebesgue spaces with general measures. The construction is also carried farther. In particular, the level function is considered as a monotone map on its natural domain, a superspace of $L^p$. These domains are shown to be Banach spaces which, although closely tied to $L^p$ spaces, are not reflexive. A related construction is given which characterizes their dual spaces.

Ograniczanie wyników

1 Formalized Mathematics

1 Studia Mathematica

1 Nakasho K.

1 Narita K.

1 Shidama Y.

1 Sinnamon G.

1 2017

1 1994

Wyniki wyszukiwania

F. Riesz Theorem

Spaces defined by the level function and their duals