By A. I. Markushevich

The 1st English variation of this marvelous textbook, translated from Russian, was once released in 3 significant volumes. during this moment English version all 3 volumes were prepare with a brand new, mixed index and bibliography. a few corrections and revisions were made within the textual content, basically in quantity II. Volumes II and III comprise a variety of references to the sooner volumes, in order that the reader is reminded of the precise statements (and proofs) of the extra ordinary effects made use of. The three-volume-in-one layout makes it effortless to turn again the pages, refresh one's reminiscence, and continue. The proofs selected are those who provide the coed the simplest 'feel' for the topic. The watchword is readability and ease. the writer was once a number one Soviet function-theorist. it truly is seldom that a professional of his stature places himself so completely on the provider of the coed. This booklet comprises over one hundred fifty illustrations and seven-hundred workouts.

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**Example text**

Then diam(A) ≤ diam(B). 4, sup{d(c, a) c, a ∈ A} ≤ sup{d(e, b) e, b ∈ B}, which is precisely the inequality that is required. 4 Suppose S is a subset of R. Then diam(S) = sup S − inf S. 1). Second, if sup S = ∞, pick b ∈ S. Then, for each p ∈ R+ , there exists some a ∈ S with a > b + p, so that diam(S) > p. Since p is arbitrary in R+ , we have diam(S) = ∞ = sup S − inf S. By a similar argument, the proposition is true if inf S = −∞. Finally, suppose that inf S and sup S are both real. Let r ∈ R+ .

10 Every set can be endowed with a metric. Can every linear space be endowed with a norm? 1) implies existence of a norm. Suppose V is a linear space and S is a basis for V . 1). The minimality of S as a spanning set for V ensures that the representation of each vector v as a sum s∈S λv,s s, where only a ﬁnite number of the λv,s are non-zero, is unique. Set ||v|| = max{|λv,s | s ∈ S}. Using the triangle inequality for the modulus function, it is easy to show that this yields a norm on V . Summary In this chapter, we have deﬁned the terms metric and metric space and given various examples.

A line in R2 is a particular set of points and the reader will no doubt recall that the distance from a point z ∈ R2 to a line x ∈ R2 ax1 + bx2 + c = 0 is given by √ the formula |az1 + bz2 + c| / a2 + b2 . z Is the supremum of a subset of R necessarily zero distance from the subset itself? Recall that the supremum is deﬁned in terms of the order, not of the metric. 3). 3) leads to an aﬃrmative answer. 5 Suppose S is a subset of R and z ∈ R. Then (i) dist(z , S) ≤ |z − sup S| with equality if z ≥ sup S; 24 2.