By Robert E. Megginson

Getting ready scholars for additional examine of either the classical works and present examine, this is often an obtainable textual content for college kids who've had a direction in actual and intricate research and comprehend the fundamental homes of L p areas. it's sprinkled liberally with examples, historic notes, citations, and unique resources, and over 450 routines offer perform within the use of the consequences constructed within the textual content via supplementary examples and counterexamples.

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Then the collection of all scalar-valued continuous functions on K is a vector space under the operations given by the usual addition of functions and multiplication of functions by scalars. Notice that if K is empty, then this vector space has the single element represented by K x IF; see the reference cited in the preceding example. For each member f of this vector space, let ll/lloo = {max{ lf(x)J : x 0 E K} ~f K =f. 0; tf K = 0. It is easy to check that this defines a norm on the vector space.

A little more work shows that the Heine-Borel property actually characterizes the finite-dimensional normed spaces among all normed spaces. 22 Lemma. If X is an infinite-dimensional normed space, then there is a linearly independent sequence (xn) in Sx such that llxn- Xmll ~ 1 whenever n =/= m. PROOF. The sequence (xn) is constructed inductively, beginning with an arbitrary element x 1 of Sx. Suppose that n ~ 2 and that linearly independent elements x1, ... xi -xkll ~ 1 if j, k :5 n -1 and j =/= k.

Suppose conversely that there are positive real numbers 8 and t such that 811xll 5 IITxll 5 tllxll for each x in X. Then the second half of the inequality shows that T is bounded while the first half shows that Tx :f. 0 whenever x :f. 0, that is, that T is one-to-one. Since IIT- 1 (Tx)ll = llxll 5 8- 1 11Txll for each x in X, the operator T- 1 is bounded, so T is an isomorphism. · Part (b) follows immediately from (a). For (c), suppose that X is a Banach space and T is an isomorphism. Let (Yn) be a Cauchy sequence in T(X).

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An Introduction to Banach Space Theory by Robert E. Megginson
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