MATH501_HW8 - Serge Ballif MATH 501 Homework 8 1 Let(n be a...

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Serge Ballif MATH 501 Homework 8 November 30, 2007 1. Let ( ϕ n ) be a Dirac sequence such that each ϕ n is continuous and supp ϕ n = B 1 /n (0) . (a) Show that ( ϕ n ( x )) converges to 0 for almost every x d . Deduce that ( ϕ n ) does not converge in L 1 ( d ) . (b) Deduce that L 1 ( d ) does not have a unity, that is, there is no g L 1 such that f * g = f for all f L 1 ( d ) . Recall that a Dirac sequence of functions ϕ n must satisfy ϕ n 0 and R ϕ n = 1. (a) For x 0 6 = 0, there is an N such that for n > N , x 0 / B 1 /n (0). Hence ϕ n ( x 0 ) = 0 for n > N . Therefore, ( ϕ n ( x )) converges to 0 for all x except x = 0. If there exists a function g L 1 such that ϕ n g a.e., then by the DCT 1 = lim n →∞ R ϕ n = R lim n →∞ ϕ n = 0. Hence, for every g L 1 , there exists n such that g ( x ) < ϕ n ( x ) for all x in a set of positive measure. Thus, ( ϕ n ) does not converge in L 1 . (b) Recall that for f in L 1 , f * ϕ n f as n → ∞ . Seeking a contradiction we suppose that L 1 ( d ) has a unity g , that is, a function g L 1 such that g * f = f for all f L 1 . Then g * ϕ n = ϕ n for all n . Moreover, g * ϕ n = ϕ n g in L 1 as n → ∞ . However, we saw that ( ϕ n ) does not converge in L 1 , so we have contradicted part (a). Having arrived at our contradiction, we must conclude that L 1 has no unity.
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