L1 Space: Difference between revisions
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===Introduction=== | ===Introduction=== | ||
From our study of integration, we know that if <math> f,g</math> are integrable functions, the following are integrable: | Let <math>(X,\mathcal{M},\mu)</math> be a measure space. From our study of integration, we know that if <math> f,g</math> are integrable functions, the following functions are also integrable: | ||
<ol> | <ol> | ||
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<li><math>cf</math>, for <math> c\in\mathbb{R}</math> | <li><math>cf</math>, for <math> c\in\mathbb{R}</math> | ||
<ol> | <ol> | ||
This shows that the set of integrable functions on any measurable space is a vector space. Furthermore, integration is a linear functional on this vector space, ie a linear function sending elements in our vector space to <math>\mathbb{R}</math>. | |||
==References== | ==References== | ||
Revision as of 08:22, 15 December 2020
Introduction
Let be a measure space. From our study of integration, we know that if are integrable functions, the following functions are also integrable:
- , for
-
This shows that the set of integrable functions on any measurable space is a vector space. Furthermore, integration is a linear functional on this vector space, ie a linear function sending elements in our vector space to .
References
