2) Iwzl = Iwi - IzI (product rule)
3) Iw + zj < Iwl + IzI (triangle inequality).
Here 1) and 2) are direct and 3) is gotten by means of the Law of Cosines and the Cauchy-Schwarz inequality (cf. also 3.4.2 in Numbers [19]) as follows:
iw+zI2 = IwI2+IzI2+2(w,z) < IwI2+IzI2+2IwIIzl = (IwI+IzI)2. 13
The product rule implies the division rule:
Iw/zi = IwI/IzI for all w, z E C, z 36 0.
The following variations of the triangle inequality are often useful:
IwI > IzI - Iw - zI , Iw + zI >_ IIwI -IzII , IIwI -IzII <- Iw - zl.
Rules 1)-3) are called evaluation rules. A map I- I
: K --+ R of a
(commutative) field K into R which satisfies these rules is called a valuation on K; a field together with a valuation in called a valued field Thus R and C are valued fields.
From the Cauchy-Schwarz inequality it follows that
-1 <
(u' z) < 1 for all w, Z E C.IwIIzI
According to (non-trivial) results of calculus, for each w, z E C" there- fore a unique real number gyp, with 0 < W < ir, exists satisfying
coscp = (w, z);
IwIIzI
W is called the angle between w and z, symbolically L(w, z) = W.
Because (w, z) = IwI IzI cos w and cos <p = - cos 0 (due to' + V = it see the accompanying figure), the Law of Cosines can be written in the form
Iw + z12 = Iw12 + Iz12 - 2Iwllzl cos-G, familiar from elementary geometry.
With the help of the absolute value of complex numbers and the fact that every non-negative real number r has a non-negative square-root V /r-1 square-roots of any complex number can be exhibited. Direct verification confirms that
for a, b E R and c := a + ib the number
+iq
2(IcI -a) ,with rl := ±1 so chosen that b = satisfies t2 = c.
Zeros of arbitrary quadratic polynomials z2 + cz + d E C[z] are now determined by transforming into a "pure" polynomial (z + Zc)2 + d - 4c2 (that is, by completing the square). Not until 9.1.1 will we show that every non-constant complex polynomial has zeros in C (the Fundamental Theorem of Algebra); for more on the problem of solvability of complex equations, compare also Chapter 3.3.5 and Chapter 4 of Numbers [19].
4. Angle-preserving mappings. In the function theory of RIEMANN, angle-preserving mappings play an important role. In preparation for the considerations of Chapter 2.1, we look at K-linear injective (consequently also bijective) mappings T : C -' C. We write simply Tz instead of T(z).
We call T angle-preserving if
IwIIzI (Tw,Tz) = ITwIITzI (w, z) for all w, z E C.
The terminology is justified by rephrasing this equality in the previously introduced language of the angle between two vectors. So translated, it says that d(Tw,Tz) = L(w,z) for all w,z E C". Angle-preserving mappings admit a simple characterization.
Lemma. The following statements about an R-linear map T : C -. C are equivalent:
i) T is
angle-preserving.ii) There exists an a E C" such that either Tz = za for all z E C or Tz=az for all z E C.
iii) There exists a number s > 0 such that (Tw, Tz) = s(w, z) for all w, z E C.
Proof i) ii) Because T is injective, a:= T1 E C". For b:= a-'Ti E C it then follows that
0 = (i, 1) = (Ti, = (ab, a) = Ial2Kb,
that is, b is purely imaginary: b = ir, r E R. We see that Tz = TI x + Ti y = a(x + iry) and so (Ti, = (a, a(x + iry)) = IaI2x. Therefore, on account of the angle-preserving character of T (take w := 1 in the defining equation), it follows that for all z E C
Ix+iyIlal2x= IlUIzl(T1,Tz) = IT1IITzl(l,z) = IaIIa(x+iry)lx, that is, Ix + iryl = Ix + iyI for all z with x 54 0. This implies that r = ±1 and we get Tz = a(x ± iy), that is, Tz = az for all z or Tz = az for all z.
ii) iii) Because (aw, az) = Ia12 (w, z) and (w, -z) = (w, z), in either case (Tw,Tz) = s(w, z) holds with s := Ial2 > 0.
iii) i) Because ITzI = v' Izl for all z, T is injective; furthermore this equality and that in iii) give
I wlI zI (Tw,Tz) = Iwllzls(w, z) = ITwIITzl (w, z). 0 The lemma just proved will be applied in 2.1.1 to the R-linear differential of a real-differentiable mapping.
In the theory of the euclidean vector spaces, a linear self-mapping T : V - V of a vector space V with euclidean scalar product ( , ) is called a similarity if there is a real number r > 0 such that ITvI = rJvI holds for all v E V; the number r is called the similarity constant or the dilation factor of T. (In case r = 1, T is called length-preserving = isometric, or an orthogonal transformation.) Because of the Law of Cosines, a similarity then also satisfies
(Tv,Tv') = r2(v,v') for all v,v' E V.
Every similarity is angle-preserving, that is, L(Tv,Tv') = L(v,v'), if one again defines L(v,v') as the value in (0,x] of the arccosine of IvI - IIv'J - 1(v,v') (and the latter one can do because the Cauchy-Schwarz inequality is valid in every euclidean space).
Above we showed that conversely in the special case V = C every angle-preserving (linear) mapping is a similarity. Actually this converse prevails in every finite-dimensional euclidean space, a fact usually proved in linear algebra courses.
Exercises
Exercise 1. Let T(z) := Az + µz, A, ti E C. Show that
a) T is bijective exactly when as -A µµ. Hint: You don't necessarily have to show that T has determinant
a_s
- µµ.
b) T is isometric, i.e., IT(z)I = IzI for all z E C, precisely when Aµ = 0 and lei + µl = 1.
Exercise 2. Let al, ... , an, bl,... , bn E C and satisfy Ev aL _ Fv=1 for all j E N. Show that there is a permutation it of 11, 2, ... , n} such that
for all vE {1,2,...,n}.
Exercise 3. For n > 1 consider real numbers ca > cl >
> c > 0.
Prove that the polynomial p(z) := co + c1z + + cnzn in C has no zero whose modulus does not exceed 1. Hint: Consider (1 - z)p(z) and note (i.e., prove) that for w, z E C with w 96 0 the equality Iw - zI = IIwl - IzMI holds exactly when z = Aw for some A > 0.
Exercise 4. a) Show that from (1 + Ivl2)u = (1 + lul2)v, u, v E C, it follows that either u = v or uv = 1.
b) Show that for u,v E C with Jul < 1, Ivi < 1 and uv # uv, we always have
I(1 + IUI2)v - (1 + Ivl2)ul > Iuv - uvl.
c) Show that for a, b, c, d E C with Ial = Ibi = Icl the complex number (a - b)(c - d) (a - d)(c - b) + i(cc - dd)3'(c7b - ca - ab) is real.
§2 Fundamental topological concepts
Here we collect the topological language and properties which are indis- pensable for function theory (e.g., "open", "closed", "compact"). Too much topology at the beginning is harmful, but our program would fail without any topology at all. There is a quotation from R. DEDEKIND's book Was rind and was sollen die Zahlen (Vieweg, Braunschweig, 1887; English trans.
by W. W. BEMAN, Essays in the Theory of Numbers, Dover, New York, 1963) which is equally applicable to set-theoretic topology, even though the latter had not yet appeared on the scene in Dedekind's time: "Die grofiten and fruchtbarsten Fortschritte in der Mathematik and anderen Wissenschaften sind vorzugsweise durch die Schopfung and Einfiihrung neuer Begriffe gemacht, nachdem die haufige Wiederkehr zusammengesetz- ter Erscheinungen, welche von den alten Begriffen nur mahselig beherrscht werden, dazu gedrangt hat (The greatest and most fruitful progress in mathematics and other sciences is made through the creation and intro- duction of new concepts; those to which we are impelled by the frequent recurrence of compound phenomena which are only understood with great difficulty in the older view)." Since only metric spaces ever occur in func- tion theory, we limit ourselves to them.
1. Metric spaces. The expression Iw - zI= vl'(u
measures the euclidean distance between the points w = u + iv and z x + iy in the plane C (figure below).
The function
CxC-R, (w,z)i- Iw-zl
has, by virtue of the evaluation rules of 1.3, the properties
Iw - zI > 0,
Iw-zl=Oaw=z,
lw - zl=lz - wl (symmetry) Iw - zI < Iw - w'I + Iw' - zi (triangle inequality) .If X is any set, a function
d:XxX-iIR, (x,y)id(x,y)
is called a metric on X if it has the three preceding properties; that is, if for all x, y, z E X it satisfies
d(x,y) > 0, d(x,y) = 0 .* x = y, d(x,y) = d(y,x), d(x, z) < d(x,y) + d(y,z).
X together with a metric is called a metric space. In X = C, d(w, z) 1w - z] is called the euclidean metric of C.
In a metric space X with metric d the set Br(c) := {x E X : d(x,c) < r}
is called the open ball of radius r > 0 with center c E X; in the case of the euclidean metric in C the balls
Br.(c)_{zEC:Iz-cI<r}, r>0
are called open discs about c, traditionally but less precisely, circles about
c.
The unit disc B1(0) plays a distinguished role in function theory. Re- calling that the German word for "unit disc" is Einheitskreisscheibe, we
will use the notation
E := B1(0) = {z E C : ]z] < 1}.
Besides the euclidean metric the set C = R2 carries a second natural metric.
By means of the usual metric Ix - 1 , x,i E IR on R we define the maximum metric on C as
d(w, z) := max{I tw - tz1,1!'w - 2''zl}, W, z E C.
It takes only a minute to show that this really is a metric in C. The "open balls"
in this metric are the open squares [Quadrate in German] Qr(c) of center c and side-length 2r.
In function theory we work primarily with the euclidean metric, whereas in the study of functions of two real variables it is often more advantageous to use the maximum metric. Analogs of both of these metrics can be introduced into any n-dimensional real vector space IR", I < n < oo.
2. Open and closed sets. A subset U of a metric space X is called open (in X), if for every x E U there is an r > 0 such that Br(x) C U. The empty set and X itself are open. The union of arbitrarily many and the intersection of finitely many open sets are each open (proof!). The "open balls" Br(c) of X are in fact open sets.
Different metrics can determine the same system of open sets; this hap- pens, for example, with the euclidean metric and the maximum metric in C = R2 (more generally in R"). The reason is that every open disc contains an open square of the same center and vice-versa.
A set C C X is called closed (in X) if its complement X \C is open. The sets
Br(c) :_ {x E X : d(x,c) < r}
are closed and consequently we call them closed balls and in the case X = C, closed discs.
Dualizing the statements for open sets, we have that the union of finitely many and the intersection of arbitrarily many closed sets are each closed.
In particular, for every set A C X the intersection A of all the closed subsets of X which contain A is itself closed and is therefore the smallest closed subset of X which contains A; it is called the closed hull of A or the closure of A in X. Notice that A = A.
A set W C X is called a neighborhood of the set M C X, if there is an open set V with M C V C W. The reader should note that according to this definition a neighborhood is not necessarily open. But an open set is a neighborhood of each of its points and this property characterizes
"openness".
Two different points c, c', E X always have a pair of disjoint neighbor- hoods:
Be(c)f1BE(c')=0
fore:=
Zd(c,d)>0.This is the Hausdorff "separation property" (named for the German math- ematician and writer Felix HAUSDOR.F'F; born in 1868 in Breslau; from 1902 professor in Leipzig, Bonn, Greifswald, and then Bonn; his 1914 treatise Grundzuge der Mengenlehre (Veit & Comp., Leipzig) contains the founda- tions of set-theoretic topology; died by his own hand in Bonn in 1942 as a result of racial persecution; as a writer he published in his youth under the pseudonym Paul MONGRE, among other things poems and aphorisms).
3. Convergent sequences. Cluster points. Following Bourbaki we define N := {0, 1,2,3,..2,3 ..}. Let k E N. A mapping {k, k + 1, k + 2, ...} - X, n'- cn is called a sequence in X; it is briefly denoted (cn) and generally k = 0. A subsequence of (cn) is a mapping f cn, in which nl < n2:5 ...
is an infinite subset of N. A sequence (cn) is called convergent in X, if there is a point c E X such that every neighborhood of c contains almost all (that is, all but finitely many) terms cn of the sequence; such a point c
is called a limit of the sequence, in symbols
c = lim c or, more succinctly, c = lim c,,.
n-oo
Non-convergent sequences are called divergent.
The separation property ensures that every convergent sequence has exactly one limit, so that the implication c = lim cn and c' = lim cn c = c', to which our notation already commits us, does in fact obtain. Also
Every subsequence (cn,) of a convergent sequence (cn) is convergent and lim cn, = lim cn.
l oo n-oo
If d is a metric on X then c = lim cn if and only if to every e > 0 there corresponds an nE E N such that d(cn, c) < e for all n > nf; for X = C with the euclidean metric this is written in the form
Icn - cI < e, i.e., cn E BE(c), for all n >
A set M C X is closed in X exactly when M contains the limit of each convergent sequence (cn) of cn E M.
A point p E X is called a cluster point or point of accumulation of the set M C X if U fl (M \ {p}) 54 0 for every neighborhood U of p. Every neighborhood of a cluster point p of M contains infinitely many points of M and there is always a sequence (cn) in M \ {p} with limcn = p.
A subset A of a metric space X is called dense in X if every non-empty open subset of X contains points of A; this occurs exactly when A = X.
A subset A of X is certainly dense in X if every point of X is a cluster point of A and in this case every point x E X is the limit of a sequence in A (proof!).
In C the set Q + iQ of all "rational" complex numbers is dense and countable. [Recall that a set is called countable if it is the image of N under some map.]