# On L 1 Space Formed by Real-Valued Partial Functions

Formalized Mathematics (2008)

• Volume: 16, Issue: 4, page 361-369
• ISSN: 1426-2630

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## Abstract

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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

## How to cite

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Yasushige Watase, Noboru Endou, and Yasunari Shidama. " On L 1 Space Formed by Real-Valued Partial Functions ." Formalized Mathematics 16.4 (2008): 361-369. <http://eudml.org/doc/266644>.

@article{YasushigeWatase2008,
abstract = {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},
author = {Yasushige Watase, Noboru Endou, Yasunari Shidama},
journal = {Formalized Mathematics},
language = {eng},
number = {4},
pages = {361-369},
title = { On L 1 Space Formed by Real-Valued Partial Functions },
url = {http://eudml.org/doc/266644},
volume = {16},
year = {2008},
}

TY - JOUR
AU - Yasushige Watase
AU - Noboru Endou
AU - Yasunari Shidama
TI - On L 1 Space Formed by Real-Valued Partial Functions
JO - Formalized Mathematics
PY - 2008
VL - 16
IS - 4
SP - 361
EP - 369
AB - 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
LA - eng
UR - http://eudml.org/doc/266644
ER -

## References

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