On L 1 Space Formed by Real-Valued Partial Functions

Yasushige Watase ; Noboru Endou ; Yasunari Shidama

Formalized Mathematics, Tome 16 (2008), p. 361-369 / Harvested from The Polish Digital Mathematics Library

Résumé

This article contains some definitions and properties refering to function spaces formed by partial functions defined over a measurable space. We formalized a function space, the so-called L1 space and proved that the space turns out to be a normed space. The formalization of a real function space was given in [16]. The set of all function forms additive group. Here addition is defined by point-wise addition of two functions. However it is not true for partial functions. The set of partial functions does not form an additive group due to lack of right zeroed condition. Therefore, firstly we introduced a kind of a quasi-linear space, then, we introduced the definition of an equivalent relation of two functions which are almost everywhere equal (=a.e.), thirdly we formalized a linear space by taking the quotient of a quasi-linear space by the relation (=a.e.).MML identifier: LPSPACE1, version: 7.9.03 4.108.1028

Détail

Publié le : 2008-01-01

Zbl 1283.46024

EUDML-ID : urn:eudml:doc:266644

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