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P. 71
Measure Theory and Functional Analysis
Notes L -Spaces: The class of all measurable functions f (x) is known as L -spaces over [a, b], if Lebesgue
P
p
– integrable over [a, b] for each p exists, 0 < p < , i.e.
b
p
|f| dx , (p 0)
a
and is denoted by L [a, b].
p
p
p-norm: The p-norm of any f L [a, b], denoted by f , is defined as
p
1
b p
f = |f| p , 0 < p < .
p
a
5.4 Review Questions
1. If f and g are non-negative measurable functions, then show that in Hölder’s inequality,
p
q
equality occurs iff some constants s and t (not both zero) such that sf + tg = 0.
2. State and prove Hölder’s Inequality.
5.5 Further Readings
Books G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, Cambridge University Press,
(1934)
L.P. Kuptsov, Hölder inequality, Springer (2001)
Kenneth Kuttler, An Introduction of Linear Algebra, BRIGHAM Young University,
2007
Online links www.m–hiKari.com
www.math.Ksu.edu
www.tandfonline.com
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