By Hille E., Tamarkin J. D.

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**Extra resources for On the Summability of Fourier Series. Third Note**

**Example text**

This topology is called the relative topology or the topology induced by τ on Y. When Y ⊂ X is equipped with its relative topology, we call Y a (topological) subspace of X. A set in τY is called (relatively) open in Y. For example, since X ∈ τ and Y ∩ X = Y, then Y is relatively open in itself. Note that the relatively closed subsets of Y are of the form Y \ (Y ∩ V) = Y \ V = Y ∩ (X \ V), where V ∈ τ. That is, the relatively closed subsets of Y are the restrictions of the closed subsets of X to Y.

By Zorn’s lemma, X has a maximal element, say (x, f ). We now leave it as an exercise to you to verify that x = ω1 and that f (ω1 ) = ω1 . You should also notice that f is uniquely determined and, in fact, f (x) is the ﬁrst element of the set Ω \ { f (y) : y < x}. In the next chapter we make use of the following result. 15 Interlacing Lemma Suppose {xn } and {yn } are interlaced sequences in Ω0 . That is, xn ≤ yn ≤ xn+1 for all n. Then both sequences have the same least upper bound in Ω0 . 14 (6), each sequence has a least upper bound in Ω0 .

Given a chain B in C, the family {A : A ∈ G for some G ∈ B} is a ﬁlter that is an upper bound for B in C. 7 are satisﬁed, so C has a maximal element. Note that every maximal element of C is an ultraﬁlter including F. For the last part, note that if X is an inﬁnite set, then F = {A ⊂ X : Ac is ﬁnite} is a free ﬁlter. Any ultraﬁlter that includes F is a free ultraﬁlter. Several useful properties of ultraﬁlters are included in the next three lemmas. 20 Lemma Every ﬁxed ultraﬁlter on a set X is of the form U x = {A ⊂ X : x ∈ A} for a unique x ∈ X.