Now we reason as follows. Since L 1~~m)1 = 1, we have 1~~m)1 ~ 1. Hence for each fixed j the sequence I is linear. We prove that I is bounded and has norm IIIII = b - a. In fact, writing J = [a, b] and remembering the norm on C[a, b], we obtain

Norm. The norm 11·11: X ~ R on a normed space (X, 11·11) is a functional on X which is not linear. 2.8-5 Dot product. The familiar dot product with one factor kept fixed defines a functional I: R3 ~ R by means of is compatible with 11-11, and II· liz. This norm is often called the natural norm defined by 11·lb and II· lb· If we choose Ilxlb = max IgJ I and IIYIIz = max l'I)jl, show that the natural norm is J J n m,n>N). This shows that (Xl(tO), X2(tO)'· .. ) is a Cauchy sequence of real numbers. Since R is complete (cf. 1.4-4), the sequence converges, say, Vector Space 50 Normed Space. Banach Space 58 Further Properties of Normed Spaces 67 Finite Dimensional Normed Spaces and Subspaces 72 Compactness and Finite Dimension 77 Linear Operators 82 Bounded and Continuous Linear Operators 91 Linear Functionals 103 Linear Operators and Functionals on Finite Dimensional Spaces 111 2.10 Normed Spaces of Operators. Dual Space 117 Hence if M is dense in X, then every ball in X, no matter how small, will contain points of M; or, in other words, in this case there is no point x E X which has a neighborhood that does not contain points of M. We shall see later that separable metric spaces are somewhat simpler than nonseparable ones. For the time being, let us consider some important examples of separable and nonseparable spaces, so that we may become familiar with these basic concepts.Spectral Properties of Bounded Self-Adjoint Linear Operators 460 9.2 Further Spectral Properties of Bounded Self-Adjoint Linear Operators 465 9.3 Positive Operators 469 9.4 Square Roots of a Positive Operator 476 9.5 Projection Operators 480 9.6 Further Properties of Projections 486 9.7 Spectral Family 492 9.8 Spectral Family of a Bounded Self-Adjoint Linear Operator 497 9.9 Spectral Representation of Bounded Self-Adjoint Linear Operators 505 9.10 Extension of the Spectral Theorem to Continuous Functions 512 9.11 Properties of the Spectral Family of a Bounded SelfAd,ioint Linear Operator 516

Metric Space 2 Further Examples of Metric Spaces 9 Open Set, Closed Set, Neighborhood 17 Convergence, Cauchy Sequence, Completeness 25 Examples. Completeness Proofs 32 Completion of Metric Spaces 41 The reader will notice that in these cases (Examples 1.5-1 to 1.5-5) we get help from the completeness of the real line or the complex plane (Theorem 1.4-4). This is typical. Examples 1.5-1 Completeness of R n and C n • space C n are complete. (Cf. 1.1-5.) F. Riesz's Lemma. Let Y and Z be subspaces of a normed space X (of any dimension), and suppose that Y is closed and is a proper subset of Z. Then for every real number (J in the interval (0,1) there is a Z E Z such thati) Construct an element x (to be used as a limit). (ii) Prove that x is in the space considered. (iii) Prove convergence Xn ~ x (in the sense of the metric).