On orthogonally additive injections and surjections

vol. 55, no. 2 (2015), 157–162

On orthogonally additive injections and surjections

Karol Baron

Summary. Let E be a real inner product space of dimension at least 2 and V a linear topological Hausdorff space. If card E ⩽ card V , then the set of all orthogonally additive injections mapping E into V is dense in the space of all orthogonally additive functions from E into V with the Tychonoff topology. If card V ⩽ card E, then the set of all orthogonally additive surjections mapping E into V is dense in the space of all orthogonally additive functions from E into V with the Tychonoff topology.

Keywords

orthogonal additivity;

inner product space;

linear topological space;

Tychonoff topology;

dense set

MSC 2010

39B55; 46C99; 46A99 Received: 2016-02-06, Accepted: 2016-04-01

Dedicated to Professor Henryk Hudzik on his 70th birthday.

1. Introduction

Let E be a real inner product space of dimension at least 2 and V a linear topological Hausdorff space.

A function f mapping E into V is called orthogonally additive, if f(x + y) = f (x) + f (y) for all x, y ∈ E with x ⊥ y.

It is well known (see [6, Corollary 10] and [4, Theorem 1]) that every orthogonally additive function f defined on E has the form

f(x) = a(∥x∥²) + b(x) for x ∈ E, (1)

Karol Baron, Institute of Mathematics University of Silesia, Bankowa 14, 40–007 Katowice, Poland (e-mail:baron@us.edu.pl)

DOI 10.14708/cm.v55i2.1109 © 2015Polish Mathematical Society

where a and b are additive functions uniquely determined by f . Consequently, we have an operator Λ which to any orthogonally additive f∶ E → V assigns a pair (a, b) of additive functions such that (1) holds, i.e.

Λ f = (a, b) (2)

where

a∶ R → V, b∶ E → V are additive and (1) holds. (3) Putting

Hom_–(E, V) = { f ∶ E → V ∶ f is orthogonally additive}

and

Hom(S, V) = { f ∶ S → V ∶ f is additive}

for S ∈ {R, E}, we see that Λ∶ Hom–(E, V) → Hom(R, V) × Hom(E, V), given by (2) and (3), is a linear bijection.

Given a non-empty set S, consider the set V^S of all functions from S into V with the usual Tychonoff topology; clearly V^Sis a linear topological space. In what follows, we consider Hom_–(E, V) and Hom(S, V) for S ∈ {R, E} with the topology induced by V^E and V^S, respectively. According to [3, Theorem 1]:

1. The isomorphism Λ∶ Hom–(E, V) → Hom(R, V) × Hom(E, V) given by (2) and (3) is a homeomorphism.

As an immediate consequence of [2, Theorem] and [3, Corollary 1] we have the follo- wing information:

2. The set

{ f ∈ Hom–(E, V)∶ f is injective and f (E) = V}

is nowhere dense in Hom_–(E, V).

Basing on the continuity of Λ⁻¹and ideas from [1, Remarks 1 and 2], we are going to show that each of the sets

{ f ∈ Hom–(E, V) ∶ f is injective}, (4)

{ f ∈ Hom–(E, V) ∶ f (E) = V} (5)

is dense in Hom_–(E, V) (cardinalities of E and V permitting).

2. Results

They read as follows.

2.1. Theorem. If card E ⩽ card V , then set (4) is dense in Hom_–(E, V).

2.2. Theorem. If card V ⩽ card E, then set (5) is dense in Hom_–(E, V).

Since the intersection of a dense set with the complement of a nowhere dense set is dense, the above theorems imply the following corollaries.

2.3. Corollary. If card E ⩽ card V , then the set

{ f ∈ Hom–(E, V) ∶ f is injective and f (E) /= V}

is dense in Hom_–(E, V).

2.4. Corollary. If card V ⩽ card E, then the set

{ f ∈ Hom–(E, V) ∶ f (E) = V and f is not injective}

is dense in Hom_–(E, V).

3. Proofs

We start with two lemmas.

3.1. Lemma. If card E ⩽ card V , then the set

{(a, b) ∈ Hom(R, V) × Hom(E, V) ∶ a(R) ∩ b(E) = {0} = b⁻¹({0})} (6)

is dense in Hom(R, V) × Hom(E, V).

To formulate the second lemma, denote byL the set of all one-dimensional linear subspaces of E and byV the set of all pairs(V¹, V2) of linear subspaces of V over the field Q of all rationals such that V¹is finite-dimensional and V1⊕ V²= V.

Observe that if L∈L^{, then E}= L ⊕ L^⊥, whence card E= c⋅ card L^⊥, and this implies (see, e.g., [7, formula (2.1) on p. 414]) that

card E= card L^⊥. (7)

3.2. Lemma. If card V ⩽ card E, then the set

⋃

L∈L ,(V1, V2)∈V{(a, b) ∈ Hom(R, V) × Hom(E, V) ∶ a([0, ∞)) = V¹, V2⊂ b(L^⊥), b(L) = {0}}

(8)

is dense in Hom(R, V) × Hom(E, V).

Proof of the Lemmas. We divide it into three parts.

Part I. It concerns both lemmas. Fix non-empty open sets U ⊂ Hom(R, V) and V ⊂ Hom(E, V). To show that U × V intersects the considered set (i.e. (6) or (8)), we may assume that V /= {0},

U = a⁰+ {a ∈ Hom(R, V) ∶ a(αⁿ) ∈ U for n ∈ {1, . . . , N}}

and

V = b⁰+ {b ∈ Hom(E, V) ∶ b(xⁿ) ∈ U for n ∈ {1, . . . , N}}

with a₀ ∈ Hom(R, V), b0 ∈ Hom(E, V), a neighbourhood U of zero in V and some α₁, . . . , αN ∈ R, x1, . . . , xN∈ E, N ∈ N.

Let H_R⊂ (0, ∞) and H^Ebe bases of the vector spacesR and E, respectively, both of them overQ. There are finite sets H_R⁰ ⊂ HRand H⁰_E ⊂ H^E such that αn ∈ LinQH⁰

Rand xn ∈ LinQH_E⁰ for n ∈ {1, . . . , N}. For every n ∈ {1, . . . , N}, α ∈ H⁰_Rand x ∈ H⁰Elet ρ⁽ⁿ⁾_α and r⁽ⁿ⁾_x be rationals such that

αn= ∑

α∈H⁰_R

ρ⁽ⁿ⁾α α , xn= ∑

x∈H⁰_E

r⁽ⁿ⁾x x . (9)

Choose now a neighbourhood U₀of zero in V such that

α∈H∑⁰

R

ρ⁽ⁿ⁾_α U₀∪ ∑

x∈H⁰

E

r⁽ⁿ⁾_x U₀ ⊂ U for n∈ {1, . . . , N}, (10)

and injective functions a1∶ H_R⁰ → V, b¹∶ H⁰E→ V such that

a1(H_R⁰) ∪ b¹(H⁰E) is linearly independent over Q, a¹(H_R⁰) ∩ b¹(HE⁰) = ∅ and

a₁(α) ∈ U⁰+ a⁰(α) for α ∈ H_R⁰, b₁(x) ∈ U⁰+ b⁰(x) for x ∈ H⁰E. (11) Part II. It concerns Lemma3.1. Let H be a basis of V overQ containing a¹(H⁰_R) ∪ b¹(H⁰E).

Since (see, e.g., the proof of [5, Lemma 4.2.2]) card(H^E/H⁰E) = card H^E = card Lin

Q

HE= card E

⩽ card V = card Lin

Q

H= card H = card(H/(a1(H⁰_R) ∪ b1(H⁰E))), we may extend b₁to an injective b₂∶ HE → H/a1(H⁰_R). Let a2∶ HR→ a1(H⁰_R) be an exten- sion of a₁and consider the additive functions a∶ R → V and b∶ E → V such that a∣^H_R= a2

and b∣^H_E = b². Clearly, b is injective, a(R) ∩ b(E) = Lin

Q

a(HR) ∩ Lin

Q

b(H^E)

= Lin

Q

a₂(HR) ∩ Lin

Q

b₂(HE) ⊂ Lin

Q

a₁(H_R⁰) ∩ Lin

Q (H/a1(H⁰_R)) = {0},

and, according to (9), (11) and (10), for any n∈ {1, . . . , N} we have a(αⁿ) − a⁰(αⁿ) = ∑

α∈H⁰

R

ρ⁽ⁿ⁾_α (a¹(α) − a⁰(α)) ∈ ∑

α∈H⁰

R

ρ⁽ⁿ⁾_α U0⊂ U (12)

and

b(xn) − b0(xn) = ∑

x∈H⁰

E

r⁽ⁿ⁾_x (b1(x) − b0(x)) ∈ ∑

x∈H⁰

E

r⁽ⁿ⁾_x U₀ ⊂ U. (13) It shows that(a, b) is in the set (6) and inU × V.

Part III. It concerns Lemma3.2. Let L be a one-dimensional linear subspace of E such that L∩ Lin

Q

H⁰_E = {0}.

Let H_Lbe a basis of L overQ, and H^L⊥a basis of L^⊥overQ. Put V1= Lin

Q

a1(H_R⁰)

and let a∶ R → V¹be an additive extension of a1such that a([0, ∞)) = V¹. Consider now a basis H of V overQ containing a¹(H_R⁰). Then, by (7),

card(H/a¹(H⁰_R)) = card H = card V ⩽ card E = card L^⊥= card H^L⊥= card(H^L⊥/HE⁰), and so there is an additive extension b∶ E → V of b1such that

b(H^L) = {0} and b(H^L⊥/HE⁰) = H/a¹(H_R⁰).

Putting

V₂= Lin

Q (H/a1(H_R⁰)),

we see that(V¹, V₂) ∈V. It shows that(a, b) is in the set (8). Moreover, by (9), (11) and (10), for every n∈ {1, . . . , N} we have (12) and (13) and so(a, b) is also in U × V.

Proof of Theorem2.1. Denote the set (6) byD. Since Λ⁻¹ is continuous, it follows from Lemma3.1that Λ⁻¹(D) is dense in Hom–(E, V), and it is enough to show that any f in Λ⁻¹(D) is injective. Indeed, let f ∈ Λ⁻¹(D). Then

(a, b) ∶= Λ f ∈ D, (14)

and if f(x) = f (y) for some x, y ∈ E, then

a(∥x∥²) − a(∥y∥²) = b(y) − b(x).

The left-hand side belongs to a(R) and the right-hand side is in b(E), whence b(x) = b(y) and x = y.

Proof of Theorem2.2. Denoting now byD the set (8), it is enough to show that any f in Λ⁻¹(D) maps E onto V. Indeed, let f ∈ Λ⁻¹(D). Keeping (14), we can find L∈ L ^and (V1, V₂) ∈V ^{such that}

a([0, ∞)) = V1, V₂⊂ b(L^⊥) and b(L) = {0}.

Fix arbitrary v∈ V. Then v = v¹+v²with v1∈ V¹, v2∈ V², and v2= b(x²) for some x²∈ L^⊥. Choose α∈ [0, ∞) with

a(α) = v¹− a(∥x²∥²) and then x1∈ L with ∥x¹∥²= α. We have

f(x¹+ x²) = a(∥x¹∥²+ ∥x²∥²) + b(x¹+ x²) = a(α) + a(∥x²∥²) + b(x²) = v¹+ v²= v, which ends the proof.

The reader interested in further problems connected with orthogonal additivity is referred to a survey paper [8] by Justyna Sikorska.

Acknowledgement The research was supported by the Silesian University Mathematics

Department (Iterative Functional Equations and Real Analysis program).

References

[1] K. Baron, On some orthogonally additive functions on inner product spaces, Ann. Univ. Sci. Budapest. Sect.

Comput. 40 (2013), 123–127.

[2] K. Baron, Orthogonally additive bijections are additive, Aequationes Math. 89 (2015), 297–299, DOI 10.1007/s00010-013-0242-6.

[3] K. Baron, On the continuous dependence of solutions to orthogonal additivity problem on given functions, Ann. Math. Sil. 29 (2015), 19–23,DOI 10.1515/amsil-2015-0002.

[4] K. Baron and J. Rätz, On orthogonally additive mappings on inner product spaces, Bull. Polish Acad. Sci.

Math. 43 (1995), 187–189.

[5] M. Kuczma, An introduction to the theory of functional equations and inequalities. Cauchy’s equation and Jensen’s inequality, 2nd ed., Birkhäuser Verlag, Basel 2009.

[6] J. Rätz, On orthogonally additive mappings, Aequationes Math. 28 (1985), 35–49,DOI 10.1007/bf02189390.

[7] W. Sierpiński, Cardinal and ordinal numbers, Monografie Matematyczne, vol. 34, Państwowe Wydawnic- two Naukowe, Warszawa 1958.

[8] J. Sikorska, Orthogonalities and functional equations, Aequationes Math. 89 (2015), 215–277, DOI 10.1007/s00010-014-0288-0.

© 2015Polish Mathematical Society