The Lp-spaces

  • Ole Christensen
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


In this chapter we introduce and analyze an important class of Banach spaces consisting of functions, the so-called L p -spaces, where p is a parameter that specifies a particular space. First, in Section 5.1 we discuss some vector spaces and concepts that play a role in the analysis of the L p -spaces. In Section 5.2 we introduce the space L 1 (\(\mathbb R\)), and show how to make it a normed space. Integration techniques in L 1 (\(\mathbb R\)) are discussed in Section 5.3; in particular, we consider techniques for interchanging an integral and an infinite sum. For p∈]1,∞], the spaces L p (\(\mathbb R\)) are introduced in Section 5.4; their counterparts Lp(a, b) concerning functions defined on a subinterval ]a, b[ \(\mathbb R\) are discussed in Section 5.5.


Banach Space Continuous Function Vector Space Compact Support Integrable Function 
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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsTechnical University of DenmarkLyngbyDenmark

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