Kostyrko, Pavel

Institute of Mathematics AS CR, v. v. i.

Matematický časopisPavel KostyrkoIndicatrix of Banach and a Space of Continuous FunctionsMatematický časopis, Vol. 19 (1969), No. 4, 336--339Persistent URL: http://dml.cz/dmlcz/126654Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1969Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czMatematický časopis 19 (1969), No. 4 INDICATRIX OF BANACH AND A SPACE OF CONTINUOUS FUNCTIONS PAVEL KOSTYRKO, Bratislava I n paper [1] certain spaces of real functions with the Baire type metric are considered. I n the present paper we shall establish connection with pa­per [1]. We shall investigate one class of continuous functions in the space Q(T, St;t eT), determined in paper [1], in connection with the indicatrix of Banach. Let us introduce the definition of the space Q (T, St;teT): Let 0 =# T <= <= <1, oo) and let -f- oo be an accumulation point of the set T. For each t eT let St c Ei = (— oo, -f oo), where each of sets St has two elements at least. Then Q (T, St;t <=T) = X St. Hence Q(T, St;teT) is the set of real functions teT defined on T, where the point f(t) e St is the value of a function / at t. Let us define a metric for this set: Q(f, g) = 1/inf {t eT :f(t) * g(t)}, if/ 4= g , e(f,g) = 0, iff=g. By investigating functions with a bounded variation S. Banach established a function n(s,f), the so called indicatrix of Banach, which determines the number of intervals (the degenerated ones too), from which the set {t : s = f(t)} consists. I n the case, where the number of these intervals is not finite we put n(s,f) = + oo. Let us define a real function n(s,f) on E± X Q (T, St; t eT) as follows: If the number of components of the set {t : s = f(t)} (c: <(1, oo)) is equal to a finite number a, then we put n(s,f) = a. I n the reverse case we put n(s,f) = = + oo. I n the following we shall consider the space Q (T, St; t e T) with a special choice of sets T and St. We shall assume T = <1, oo) and St = S for each t eT, where S is an interval. This space we shall denote by Q. I n the following we shall investigate a subspace G of all continuous functions of the space Q. Theorem 1. Let C be a subspace of all continuous functions of the space Q. 336 Then the set B = {/e o : V V 3 3 f(u) = s ^/(*>)} seSi=l M>i v>u is a residual set in C. Proof . Let s e S. Let us put B(s) = {feC: V 3 3 f(u) = s^f(v)}. i= l M>i^>M Then evidently B(s) = O U G,(tt)> where G,(tf) = {/: 3 /(ti) = 5 -^/(v)} i=l w>i ^>w (M > 1). Lemma 1. TAe set B(s) is dense in C. P r o o f of L e m m a 1. We shall show tha t for e ach / 0 e C and e (0 < s < 1) there exists gr e i?(s) such tha t g e K(fo, e) = {/: D(/,/o) < «}• We shall define the function g as follows: g(t) = /0(£) for t ^ 2/e. Since /0(2/e) e $ there exist numbers p, q such that 5, /0(2/e) e (p, q} <= /S. Then we put gr(£) = \(p + q) + + \(q —• p) sin (t + t0) for £ > 2/e, where £0 (0 ^ to < 2n) is determined by the condition \(p + q) + \(q — p) sin (2/e + t0) = fo(2\e). This will guarantee tha t g is a continuous function in 2/e. From the construction of the function g it follows directly tha t g e B(s) and o(g,fo) < e/2 < e. Lemma 2. The set B(s) is a (?<? in C. P r o o f of L e m m a 2. First we show that the set Gs(u) = {/: 3 f(u) = v>u = s ^f(v)} (u > 1) is open in C. If g e K(f9 l\v), then K(f, l\v) c Gs(u), because for g e K(f, l\v), g(t) = f(t) (t e <1, v}) holds, hence g e C75(^). 00 The set B(s) = n U Gs(^) as an intersection of a countable family of open i=lu>i sets U C7S(̂ ) is a (?<$ in C. u">i Lemma 3. The set B(s) is residual in C. P r o o f of L e m m a 3. According to Theorem 8.4 (see p . 88) of the mono-graph [3] each Fa set, the complement of which is dense, is a set of the first category. From Lemma 1 and Lemma 2 it follows (taking complements) t ha t for each s e S the set B(s) is residual. Since (according to the assumption) the set S is an interval, then evidently the following lemma holds. Lemma 4. Let R be a countable dense subset of S, which includes min S and max S (if they exist). Then to each s e S — R there are r\, r2 e R such that n < s <r2. 337 Let us prove the statement of Theorem 1. Let us form a set B* = n B(r). As each of sets B(r) (r e B) is according to Lemma 3 residual and B is countable, then B* is residual also. Evidently it is sufficient to prove the inclusion B* <-= B. L e t / e B* and s e S — B. From Lemma 4 it follows tha t there exist numbers n,r2eB such that n < s < r2. Since feB(ri), for an arbitrary natural number i there are the numbers ui, v± (i ^ ui < Vi) such tha t f(ui) = n =>-=£f(vi). From the condition/ e 2?(r2) there follows the existence of the numbers u2, v2 (vi ^ u2 < v2) such that f(u2) = r2 ^f(v2). Then f(ui) < s <f(u2) and from the properties of the continuous functions there follows the existence u (ui < u < u2) such that f(u) = s. We have shown tha t for each s e 8 — B and for each natural i there are the numbers u and v (v = u2) such that i ^ u < < v and f(u) = s -^ f(v). If s e B, then the existence of the numbers u, v with the required qualities follows from the inclusion B* c: B(s). The Theorem is therefore completely proved. Theorem 2. L/et the space G and the function n(s, f) be given. Then the set A = {feC: V n(s,f) = + 0 0 } seS is residual in G. Proof . The statement of Theorem 2 follows from Theorem 1 and from the evident inclusion B c: A. R e m a r k . In paper [2] a theorem is proved analogical to Theorem 2, regard-ing the space (7(0, 1) of all real continuous functions defined on the interval <0, 1>. Lemma 5. The space G is complete. Proof . As the space _Q is complete (see [1], Theorem 4), it is sufficient to show tha t the set G is closed in Q. Let fn-> f (fn e C,n = 1, 2, ...) and let to E < 1, 00) be arbitrary. If for w > w 0 w e have q(fn, / ) < l/£0, then inf {t : fn(t) -^ 7^ / (0 ) > ô and there exists d > 0 such tha t for t e <1, to + d) we have f(t) = = /«o(0> i- e- ^ n e func t ion / i s continuous at every point Zo- Hence f e C. Corollary 1. Let the space C and the function n(s,f) be given. Then the set A = {feC: V w(*,/) = + 00} seS is a set of the second category in C. Proof . In a complete space C (Lemma 5) we have every residual set according to the well-known Baire Theorem (see [3], p. 80) a set of the second category. The set A, which is considered in the statement of the corollary, is according to Theorem 2 residual, consequently it is a set of the second category. In the following we shall investigate the space Q (T, St) t e T) under the 338 assumption that the set T is countable and for each t eT we have St = S, where S( <-= E\) is an arbitrary set. Let us denote this space by Qs. The function n(s,f) defined on E\ x Qs denotes evidently the number of points of the set Theorem 3. Let the space Qs and the function n(s, f) be given.Then the following statements are equivalent: (a) the set S is countable, (b) the set A = {/G QS : V n(s,f) = -f- 00} is residual in Qs. seS Proof . (a)-> (b): Let us put D = {feQs: V V 3 * = / ( * ) } . Let seS and teT. Let us put D(s, t) = {feQs:s= f(t)}. The set D(s, t) is open because itfeD(s, t), then K(f, \jt) <= D(s, t). 00 Let us put D(s) = n u D(s,t). The set D(s) is evidently a (?<? in .£?£. We shall show that D(s) is dense in Qs • For feQs and e > 0 let us define </ e Qs as follows: g(t)----f(t) tor t < 2/e, gr(j) = 5 for t > 2/e . Evidently gr G Z)(S) and g e K(f, e). According to the above mentioned Theorem of the monograph [3] the set D(s) (s e S) is residual and of such a quality is the set D = n D(s) <= A too. seS (b) -» (a): This implication wall be proved by contradiction. Let the set S be uncountable. As for an arbitrary / G QS the setf(T) is countable, A = 0 holds and the void set in the complete space Qs (see [1], Theorem 4) is not residual. Corollary 2. Let P be the space of all sequences with the Baire metric values in the set S (<= Ei). I^et A be the set of all these a = {an}™ G P, for which the set {n : an = s} is infinite for each s G S. Then the set A is residual in P if and only if the set S is countable. REFERENCES [1] K o s t y r k o P., O neKomopux npocmpancmeax c MempuKou muna Eapa, Mat.-fyz. ca-sop. 16 (1966), 143-153 . [2] Sawye r S., Some topological properties of the function n(y), Proc. Amer. Math. Soc. 18 (1967), 3 5 - 4 0 . [3] S ikorsk i R., Funkcje rzeczywiste I, Warszawa 1958. Received April 17, 1968. Katedra algebry a teorie 6isel Prirodovedeckej fakulty Univerzity Komenskeho Bratislava 3 3 9 

English

Indicatrix of Banach and a Space of Continuous Functions

Czech digital mathematics library

http://dml.cz/bitstream/handle/10338.dmlcz/126654/MathSlov_19-1969-4_16.pdf

Indicatrix of Banach and a Space of Continuous Functions

Abstract

Similar works

Full text

Available Versions

Institute of Mathematics AS CR, v. v. i.

Czech digital mathematics library