The Lp-spaces

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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