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By Anton Hemereijick, Ben Knapen, Ellen van Doorne

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We discuss some examples of a non-unitary isometry in a separable Hilbert space. (1) Let H = £2 (= £2 (1'1) ). Let {en : n E 1'1} denote the standard orthonormal basis of H (consisting of sequences with a 1 in one co-ordinate and 0 in all other co-ordinates). c(H) such that Sen = en+1 \;In E 1'1; equivalently, we have For obvious reasons, this operator is referred to as a "shift" operator; in order to distinguish it from a near relative, we shall refer to it as the unilateral shift. It should be clear that S is an isometry whose range is the proper subspace jvl = {el}1-, and consequently, S is not unitary.

6. Some topological considerations 21 We conclude this section with some exercises on the uniform boundedness principle. 17 (1) Suppose S is a subset of a (not necessarily complete) normed space X which is weakly bounded, meaning that ¢(S) is a bounded set in C, for each ¢ E X*. Then show that S is norm bounded, meaning that sup{ Ilxll : x E S} < 00. 3(2)') (2) Show that the completeness of Y is not needed for the validity of the uniform boundedness principle. ) (3) Suppose {Tn : n = 1,2, ... } c £(X, Y) is a sequence of bounded operators, where X, Yare Banach spaces, and suppose the sequence converges "strongly";' i.

Weak*) topology. Further, it follows from the definitions that if X is any normed space, then: Chapter 1. , closed). 5, for instance; but both forms of usage have become so wide-spread that this is a (minor) conflict that one has to get used to. s. X. In this case, the weak* topology on X* should be thought of as the topology induced by the family of "evaluation functionals" on X* coming from X. 8 (1) Let X be a normed space. Show that the weak topology on X is a Hausdorff topology, as is the weak * topology on X*.

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Aftershocks; Economic Crisis, Institutional Choice by Anton Hemereijick, Ben Knapen, Ellen van Doorne

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