- Email: [email protected]
On mappings approximately transferring relations in finite-dimensional normed spaces
Linear Algebra and its Applications 460 (2014) 125–135
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
On mappings approximately transferring relations in finite-dimensional normed spaces Paweł Wójcik Institute of Mathematics, Pedagogical University of Cracow, Podchor¸ ażych 2, 30-084 Kraków, Poland
a r t i c l e
i n f o
Article history: Received 6 February 2014 Accepted 29 July 2014 Available online 9 August 2014 Submitted by P. Semrl MSC: 15A86 47B49 39B82
a b s t r a c t We discuss the problem of stability of relations preserving property for finite-dimensional normed spaces. Moreover, we obtain similar results for other orthogonality relations. Next, we show a counterexample for infinite-dimensional normed spaces. © 2014 Elsevier Inc. All rights reserved.
Keywords: The Birkhoff orthogonality Isosceles orthogonality Approximate orthogonality Linear mappings orthogonality preserving Stability
1. Introduction An orthogonality preserving property can be introduced, in the most natural way, for linear mappings between inner product spaces. If X and Y are real or complex inner E-mail address: [email protected]. http://dx.doi.org/10.1016/j.laa.2014.07.046 0024-3795/© 2014 Elsevier Inc. All rights reserved.
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P. Wójcik / Linear Algebra and its Applications 460 (2014) 125–135
product spaces with the standard orthogonality relation, a linear mapping f : X → Y which satisfies the condition ∀x,y∈X
x⊥y
⇒
f (x) ⊥ f (y)
is called orthogonality preserving (o.p.). A linear orthogonality preserving mapping has to be a linear similarity (cf. [4]). Let us now introduce the notion of approximate orthogonality. For ε ∈ [0, 1), we say that vectors u, v are ε-orthogonal (u ⊥ε v) whenever |u|v| ε u · v . A linear mapping f : X → Y which satisfies ∀x,y∈X
x⊥y
⇒
f (x) ⊥ε f (y)
is called ε-approximately orthogonality preserving (ε-a.o.p.). Linear ε-a.o.p. mappings are, in a sense, approximate similarities (cf. [4]). Next, we formulate a stability problem: whether for each linear mapping f approximately orthogonality preserving, there exists a linear mapping h preserving orthogonality, which is close to f . The answer is affirmative. If f is ε-a.o.p., then there is h preserving orthogonality such that f − h (1 − 1−ε 1+ε ) f ; see [5,10]. In a normed space, one can define various orthogonality relations and one can consider linear mappings (approximately) preserving this relations. For example: the Birkhofforthogonality x ⊥B y
:⇔
∀λ∈K
x x + λy ,
and the ε-Birkhoff-orthogonality (see [7,2,3]) x ε⊥B y
:⇔
∀λ∈K
(1 − ε) x x + λy .
The above stability problem has been also carried out for ⊥B , ε⊥B (see [9]). For real normed space it can be considered the isosceles orthogonality: x ⊥i y
:⇔
x + y = x − y ,
and the ε-isosceles orthogonality x ε⊥i y
:⇔
x + y − x − y ε x + y + x − y .
For ⊥i , ε⊥i the stability problem has been solved in [6]. Of course, in an inner product space we have ⊥ = ⊥B = ⊥i . We will consider only finite-dimensional normed spaces, but we will obtain similar results for other orthogonality relations.
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2. Preliminaries Let X, Y be normed spaces, and let R1 ⊂ X × X, R2 ⊂ Y × Y be relations. We say that linear mapping f : X → Y transfers relation R1 into R2 (f is relation transferring), if it satisfies ∀x,y∈X
x R1 y
⇒
f (x) R2 f (y).
Now, we consider a family of relations R2ε ⊂ Y × Y for ε ∈ [0, 1) which satisfies the following conditions: (a)
ε ε
⇒
R2ε ⊂ R2ε ;
(b)
ε=0
⇒
R2ε = R2 .
(1)
The mapping f : X → Y , which satisfies the condition ∀x,y∈X
x R1 y
⇒
f (x) R2ε f (y)
(2)
is called ε-relation transferring. We say that a relation S ⊂ Y × Y is weakly homogeneous, if x S y implies αx S αy for all α ∈ K and for all x, y ∈ Y . A relation S ⊂ Y × Y is called homogeneous, if x S y implies αx S βy for arbitrary α, β ∈ K and all x, y ∈ Y . Next, we say that a family of relations R2ε ⊂ Y × Y for ε ∈ [0, 1) is continuous with respect to the relation R2 ⊂ Y × Y , if for each sequence (εn )n∈N such that 0 εn < 1, limn→+∞ εn = 0 and for all sequences (an )n∈N , (bn )n∈N ⊂ Y such that limn→+∞ an = a, limn→+∞ bn = b we have (∀n∈N an R2εn bn ) ⇒ a R2 b. For example, the relations ⊥, ⊥ε are weakly homogeneous and the family of relations ⊥ε for ε ∈ [0, 1) is continuous with respect to the relation ⊥. 3. General case Throughout this section, X, Y denote normed spaces (over the same field K). We are going to prove a stability of linear mappings transferring a relation. This result will be useful for further considerations. Lemma 1. Let R1 ⊂ X × X, R2ε ⊂ Y × Y be relations. Assume that R2ε is weakly homogeneous. If f : X → Y satisfies (2), then for α ∈ K the mapping αf also satisfies (2). Proof. Assume x R1 y. Then f (x) R2ε f (y). Since R2ε is weakly homogeneous, we obtain αf (x) R2ε αf (y), so αf satisfies (2). 2 Lemma 2. Let R1 ⊂ X × X, R2ε ⊂ Y × Y be relations. Suppose that a family {R2ε }ε∈[0,1) is continuous with respect to the relation R2 . Assume that limn→+∞ εn = 0 with εn ∈ [0, 1). If the mappings fn : X → Y are εn -relation transferring (for all n ∈ N) and
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satisfy limn→+∞ fn = f (pointwisely), then the mapping f : X → Y transfers relation (R1 into R2 ). Proof. Assume x R1 y. Then fn (x) R2εn fn (y). Since limn→+∞ fn = f and a family {R2ε }ε∈[0,1) is continuous with respect R2 , we get f (x) R2 f (y). Thus f preserves relation. 2 Now, we are able to prove the stability of the transferring relation property. The following theorem is the main theorem in this paper. Theorem 3. Let X, Y be finite-dimensional normed spaces and let R1 ⊂ X ×X, R2 , R2ε ⊂ Y × Y . Suppose that relations R2ε are weakly homogeneous for all ε ∈ [0, 1) and let the family of relations {R2ε }ε∈[0,1) be continuous with respect to the relation R2 . Then, for an arbitrary δ > 0, there exists ε > 0 such that for any linear ε-relation transferring mapping f : X → Y , there exists a linear mapping g: X → Y which transfers relation, such that f − g δ min f , g .
(3)
Proof. The case f = 0 is trivial. Then g := 0. Thus we may consider only nonzero linear mappings. First, we observe that (3) holds if and only if f − g δ f and f − g δ g
(4)
hold. Suppose that the assertion is not true, that there exists δ > 0 such that for each εn := n1 there exists a linear mapping εn -relation transferring fn : X → Y such that for each linear mapping g: X → Y which transfers relation, (3) is not true. Therefore from (4) we get fn − g > δ fn
or fn − g > δ g ,
whence fn g − fn fn > δ
fn g or − g g > δ.
(5)
Note that ffnn ∈ B(0, 1) ⊂ L(X; Y ) for all n ∈ N (here B(0, 1) denotes a closed unit ball). Since L(X; Y ) is finite-dimensional, the closed unit ball is compact. Therefore there f exists f ∈ L(X; Y ) and a subsequence ( fnnk )k∈N that k
lim
k→+∞
fnk = f. fnk
(6)
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f
By Lemma 1, for all k ∈ N, the mappings fnnk are εnk -relation transferring. Lemma 2 k yields that f exactly transfers the relation, and from (6) we have f = 1. Fix k ∈ N. Now we are able to substitute g by fnk · f in (5) fnk fnk fnk · f fnk · f > δ or − − fn · f fn · f > δ. fn fnk k k k Since f = 1, we have, for an arbitrary fixed k: fnk fn − f > δ, k which contradicts (6). 2 4. Orthogonality preserving property and stability problem For a normed space X (over the field K ∈ {R, C}) we consider ⊥i , ⊥εi and ⊥B , ⊥εB . It is easy to check that the both of families of relations {ε⊥i }ε∈[0,1) , {ε⊥B }ε∈[0,1) satisfy conditions (1). Moreover, ⊥i , ⊥B , ε⊥i , ε⊥B are weakly homogeneous. We say that f : X → Y preserves the i-orthogonality (or preserves the Birkhoff orthogonality), if it satisfies ∀x,y∈X
x ⊥i y
⇒
f (x) ⊥i f (y),
or
∀x,y∈X x ⊥B y ⇒ f (x) ⊥B f (y) .
Clearly f transfers relation ⊥i ⊂ X × X into ⊥i ⊂ Y × Y (or ⊥B ⊂ X × X into ⊥B ⊂ Y × Y ). Koldobsky, Blanco and Turnšek (cf. [8,1]) proved the following result, which describes the class of linear mappings preserving B-orthogonality. Theorem 4. Let f : X → Y be a linear operator. Then f preserves the Birkhoff–James orthogonality, if and only if, for some γ > 0, f (x) = γ x , x ∈ X. Now, we can define mappings, which approximately preserve the i-orthogonality (or approximately preserve the Birkhoff orthogonality). For ε ∈ [0, 1), f : X → Y is called an ε-i-orthogonality preserving mapping (or an ε-B-orthogonality preserving mapping), if it satisfies ∀x,y∈X
x ⊥i y
⇒
f (x) ε⊥i f (y),
(7)
∀x,y∈X x ⊥B y ⇒ f (x) ε⊥B f (y) .
(8)
or
Then, we may to say that f is ε-relation transferring.
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The following characterization of i-orthogonality preserving mappings was obtained in [6]. Theorem 5. Let f : X → Y be a linear operator. Then f is an ε-i-orthogonality preserving mapping, if and only if, ( 1−ε 1+ε ) f · x f x f · x , x ∈ X. Now, we obtain the results which are connected with the stability problem. Theorem 6. Let X, Y be finite-dimensional normed spaces. Then, for an arbitrary δ > 0 there exists ε > 0 such that for any linear mapping f : X → Y satisfying (7) there exists a linear mapping g: X → Y preserving i-orthogonality such that f − g δ · min f , g . Proof. Clearly relation ε⊥i is weakly homogeneous and the family {ε⊥i }ε∈[0,1) is continuous with respect to the relation ⊥i . Moreover, family {ε⊥i }ε∈[0,1) satisfies (1). Since f satisfies (7), so we can say that f is ε-relation transferring. Thus we are able to apply Theorem 3 and we obtain assertion. 2 In the same manner we can obtain the following theorem. Theorem 7. Let X, Y be finite-dimensional normed spaces. Then, for an arbitrary δ > 0, there exists ε > 0 such that for any linear ε-B-orthogonality preserving mapping f : X → Y there exists a linear B-orthogonality preserving mapping g: X → Y such that f − g δ min f , g . Theorems 6, 7 were obtained in the other way (see [6] and [9, Proposition 4.4]). But here we gave new proof and next we will obtain results for others relations. Now, for normed spaces, we can consider others notions of orthogonality. One of them is the Roberts orthogonality (R-orthogonality): x ⊥R y
:⇔
∀λ∈R
x + λy = x − λy ,
and an approximate Roberts orthogonality: x ⊥εR y
:⇔
∀λ∈R
x + λy 2 − x − λy 2 4ε x · λy .
If the norm comes from the an inner product, then ⊥ = ⊥R and ⊥ε = ⊥εR . Next, we could consider the class of linear mappings preserving R-orthogonality: ∀x,y∈X
x ⊥R y
⇒
f (x) ⊥R f (y),
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and ∀x,y∈X
x ⊥R y
⇒
f (x) ⊥εR f (y).
Similarly, for the Pythagorean orthogonality (P-orthogonality): x ⊥P y
:⇔
x + y 2 = x 2 + y 2 ,
and an approximate P-orthogonality: x ⊥εP y
:⇔
x + y 2 − x 2 − y 2 2ε x · y ,
one can consider mappings preserving P-orthogonality and their stability. Of course, in an inner product space we have ⊥ = ⊥P and ⊥ε = ⊥εP . Applying Theorem 3 we get that for R-orthogonality and P-orthogonality the stability problem has an affirmative answer for finite-dimensional normed spaces. The case of infinitely-dimensional spaces remains an open problem. 5. Linear mappings which preserve angle and stability problem For real inner product space (X, ·|·) we define the relation which is connected with the notion of angle. Fix c ∈ (−1, 1). For x, y ∈ X we define x ∠c y
:⇔
x|y = c x · y .
It means that the size of the angle between the vectors x and y equals arccos(c). We say that f : X → Y preserves the angle, if it satisfies ∀x,y∈X
x ∠c y
⇒
f (x) ∠c f (y).
(9)
It is easy to see that ∠0 = ⊥. Next, we fix ε ∈ [0, 1) and we define x ∠εc y
:⇔
x|y − c x · y ε x · y .
It means that the size of the angle satisfies c − ε arccos(c) c + ε. The linear mapping f : X → Y which satisfies the condition ∀x,y∈X
x ∠c y
⇒
f (x) ∠εc f (y)
(10)
is called approximately angle preserving. It is easy to check that the family of relations {∠εc }ε∈[0,1) satisfies conditions (1). Moreover, ∠c , ∠εc are weakly homogeneous. Therefore, using Theorem 3, we obtain the following theorem.
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Theorem 8. Let X, Y be finite-dimensional inner product spaces. Then for an arbitrary δ > 0 there exists ε > 0 such that for any linear mapping f : X → Y satisfying (10) there exists a linear mapping g: X → Y satisfying (9) such that f − g δ · min f , g . 6. Counterexample We now turn our attention to showing that for B-orthogonality the stability problem has a negative answer for infinite-dimensional normed spaces. The examples from [6] and from [9] concern the case of an infinite-dimensional domain X and an infinite-dimensional range Y . Therefore the following question appears. Does the problem of stability have a positive answer in the case of finite-dimensional domain X and infinite-dimensional range Y ? Let X, Y be normed spaces. A linear mapping f : X → Y is called an ε-isometry if it satisfies ∀x∈X
(1 − ε) x f (x) (1 + ε) x .
(11)
The following lemma will be useful. Lemma 9. Every linear ε-isometry is an ε -B-orthogonality preserving mapping, with ε = 1 − 1−ε 1+ε . Proof. Let x, y ∈ X and assume that x ⊥B y, i.e., x x + αy for all α ∈ K. Let λ ∈ K. Then from (11) we have f (x) (1 + ε) x (1 + ε) x + λy (1 + ε) 1 f (x + λy) 1−ε 1 + ε f (x) + λf (y), = 1−ε hence for any λ ∈ K we get (1 − ε ) f (x) f (x) + λf (y) and f is ε -B-orthogonality preserving. 2 Example 10. Next, we consider the space R2 with the standard inner products. We denote by H the space R2 with the Euclidean norm · H coming from this inner product. We can inscribe a regular polygon which has 2n sides inside a closed unit ball B((0, 0); 1) ⊂ H. Let Pn denote this “2n -regular polygon”. The set Pn is convex, absorbed and balanced. Hence this set introduces a new norm · n by the Minkowski functional. It is easy to see that Pn ⊂ B((0, 0); 1) and αn B((0, 0); 1) ⊂ Pn for some αn ∈ (0, 1). Obviously, αn 1, if n → +∞. So, we get αn x n x H x n . Let Xn denote the space R2 with the norm · n .
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Next, we define a space Y := y = (y1 , y2 , y3 , . . . , yn , 0, 0, . . .) : yk ∈ R2 , n ∈ N and we introduce a norm y Y := max y1 H , y2 2 , y3 3 , . . . , yn n for y = (y1 , y2 , . . . , yn , 0, 0, . . .) ∈ Y . Clearly dim H = 2, dim Y = ∞. Now, we define a linear mapping fk : H → Y by fk (x) := (0, . . . , 0, x, 0, 0, . . .), where x appears only at the kth coordinate. We will prove that fk is an εk -isometry with εk := α1k − 1 (hence εk 0). For any x ∈ H, it follows that (1 − εk ) x H x H x k
1 x H = (1 + εk ) x H . αk
Since fk (x) Y = x k we have ∀x∈X
(1 − εk ) x H fk (x)Y (1 + εk ) x H .
Let W be any normed space. By S(W ), B(W ) we denote a unit sphere and a closed unit ball, respectively. The following lemma gives us a characterization of linear isometry h: H → Y . Note that for some n ∈ N we have h(x) = h1 (x), h2 (x), h3 (x), . . . , hn (x), 0, 0, . . . , where each of the mappings h1 : H → H, hm : H → Xm (for m = 2, 3, . . . , n) is linear (the number n depends only on h). Lemma 11. If h: H → Y is linear isometry, then h1 = 1. Proof. Suppose that h1 < 1. Then there is c ∈ (0, 1) that for all x ∈ S(H) we have h1 (x) H c < 1. We define
h: H → Y by
h(x) := (0, h2 (x), h3 (x), . . . , hn (x), 0, 0, . . .). For any x ∈ S(H) we get 1 = h(x) = max h1 (x)H , h2 (x)2 , h3 (x)3 , . . . , hn (x)n = max h2 (x)2 , h3 (x)3 , . . . , hn (x)n =
h(x), and thus the mapping
h is also isometry. Now, we might write
h: H → X2 × . . . × Xn . A routine application of mathematical induction shows that the Cartesian product of finite number of simplexes is a simplex. Since a norm in X2 × . . . × Xn is given by u = max{ u2 2 , u3 3 , . . . , un n }, then it can be shown that
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B(X2 × . . . × Xn ) = B(X2 ) × . . . × B(Xn ). Each of the sets B(X2 ), . . . , B(Xn ) is a simplex. Therefore the closed unit ball in these normed spaces X2 × . . . × Xn is also a simplex. It is easy to see that
h(H) is an isometry and a set
h: H →
h(H) ∩ B(X2 × . . . × Xn )
is a closed unit ball in the space h(H). It is well-known that the intersection of twodimensional space and simplex is a simplex. Therefore the closed unit ball
h(H) ∩B(X2 × . . . × Xn ) is a simplex in the space
h(H). Thus the space
h(H) is not strictly convex. On the other hand, the space H is strictly convex, which contradicts the isometry
h(H). 2 h: H →
Further, we show that for B-orthogonality the stability problem has a negative answer. By Lemma 9, the mappings fk : H → Y are ε k -B-orthogonality preserving mapping, 1 k with ε k = 1 − 1−ε k → 0). 1+εk , where εk = αn − 1 (and ε Suppose that the problem of stability has a positive answer. Let 0 < δ < 1. Then, for ε k sufficiently small (i.e., k sufficiently large), there exists some linear mapping gk : H → Y preserving B-orthogonality such that fk − gk δ min fk , gk .
(12)
The mapping gk is a linear similarity by Theorem 4. It is easy to see that for some linear isometry hk : H → Y there is gk = gk hk . From (12) we get fk − gk δ gk .
(13)
Let h = (h1 , h2 , . . . , hn , 0, 0, . . .): H → Y be any isometry. Then there is xo ∈ S(H) such that h1 (xo ) H = 1 by Lemma 11. For all γ > 0, we obtain fk − γh fk (xo ) − γh(xo )Y = (0, . . . , 0, xo , 0, 0, . . .) − γ h1 (xo ), h2 (xo ), . . . , hn (xo ), 0, 0, . . . = max γ h1 (xo )H , . . . γ h1 (xo )H = γ. For the arbitrary mapping gk , there is an isometry hk such that gk = gk hk . If we put gk instead of γ and hk instead of h in the previous inequality, then the above consideration yields gk fk − gk . This fact is conflicting with (13). References [1] A. Blanco, A. Turnšek, On maps that preserve orthogonality in normed spaces, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 709–716. [2] J. Chmieliński, On an ε-Birkhoff orthogonality, JIPAM. J. Inequal. Pure Appl. Math. 6 (3) (2005), Art. 79. [3] J. Chmieliński, Remarks on orthogonality preserving mappings in normed spaces and some stability problems, Banach J. Math. Anal. 1 (1) (2007) 117–124.
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[4] J. Chmieliński, Linear mappings approximately preserving orthogonality, J. Math. Anal. Appl. 304 (2005) 158–169. [5] J. Chmieliński, Stability of the orthogonality preserving property in finite-dimensional inner product spaces, J. Math. Anal. Appl. 318 (2006) 433–443. [6] J. Chmieliński, P. Wójcik, Isosceles-orthogonality preserving property and its stability, Nonlinear Anal. 72 (2010) 1445–1453. [7] S.S. Dragomir, On approximation of continuous linear functionals in normed linear spaces, An. Univ. Timiş. Ser. Ştiinţ. Mat. 29 (1991) 51–58. [8] A. Koldobsky, Operators preserving orthogonality are isometries, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 835–837. [9] B. Mojškerc, A. Turnšek, Mappings approximately preserving orthogonality in normed spaces, Nonlinear Anal. 73 (2010) 3821–3831. [10] A. Turnšek, On mappings approximately preserving orthogonality, J. Math. Anal. Appl. 336 (2007) 625–631.
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
On mappings approximately transferring relations in finite-dimensional normed spaces Paweł Wójcik Institute of Mathematics, Pedagogical University of Cracow, Podchor¸ ażych 2, 30-084 Kraków, Poland
a r t i c l e
i n f o
Article history: Received 6 February 2014 Accepted 29 July 2014 Available online 9 August 2014 Submitted by P. Semrl MSC: 15A86 47B49 39B82
a b s t r a c t We discuss the problem of stability of relations preserving property for finite-dimensional normed spaces. Moreover, we obtain similar results for other orthogonality relations. Next, we show a counterexample for infinite-dimensional normed spaces. © 2014 Elsevier Inc. All rights reserved.
Keywords: The Birkhoff orthogonality Isosceles orthogonality Approximate orthogonality Linear mappings orthogonality preserving Stability
1. Introduction An orthogonality preserving property can be introduced, in the most natural way, for linear mappings between inner product spaces. If X and Y are real or complex inner E-mail address: [email protected]. http://dx.doi.org/10.1016/j.laa.2014.07.046 0024-3795/© 2014 Elsevier Inc. All rights reserved.
126
P. Wójcik / Linear Algebra and its Applications 460 (2014) 125–135
product spaces with the standard orthogonality relation, a linear mapping f : X → Y which satisfies the condition ∀x,y∈X
x⊥y
⇒
f (x) ⊥ f (y)
is called orthogonality preserving (o.p.). A linear orthogonality preserving mapping has to be a linear similarity (cf. [4]). Let us now introduce the notion of approximate orthogonality. For ε ∈ [0, 1), we say that vectors u, v are ε-orthogonal (u ⊥ε v) whenever |u|v| ε u · v . A linear mapping f : X → Y which satisfies ∀x,y∈X
x⊥y
⇒
f (x) ⊥ε f (y)
is called ε-approximately orthogonality preserving (ε-a.o.p.). Linear ε-a.o.p. mappings are, in a sense, approximate similarities (cf. [4]). Next, we formulate a stability problem: whether for each linear mapping f approximately orthogonality preserving, there exists a linear mapping h preserving orthogonality, which is close to f . The answer is affirmative. If f is ε-a.o.p., then there is h preserving orthogonality such that f − h (1 − 1−ε 1+ε ) f ; see [5,10]. In a normed space, one can define various orthogonality relations and one can consider linear mappings (approximately) preserving this relations. For example: the Birkhofforthogonality x ⊥B y
:⇔
∀λ∈K
x x + λy ,
and the ε-Birkhoff-orthogonality (see [7,2,3]) x ε⊥B y
:⇔
∀λ∈K
(1 − ε) x x + λy .
The above stability problem has been also carried out for ⊥B , ε⊥B (see [9]). For real normed space it can be considered the isosceles orthogonality: x ⊥i y
:⇔
x + y = x − y ,
and the ε-isosceles orthogonality x ε⊥i y
:⇔
x + y − x − y ε x + y + x − y .
For ⊥i , ε⊥i the stability problem has been solved in [6]. Of course, in an inner product space we have ⊥ = ⊥B = ⊥i . We will consider only finite-dimensional normed spaces, but we will obtain similar results for other orthogonality relations.
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127
2. Preliminaries Let X, Y be normed spaces, and let R1 ⊂ X × X, R2 ⊂ Y × Y be relations. We say that linear mapping f : X → Y transfers relation R1 into R2 (f is relation transferring), if it satisfies ∀x,y∈X
x R1 y
⇒
f (x) R2 f (y).
Now, we consider a family of relations R2ε ⊂ Y × Y for ε ∈ [0, 1) which satisfies the following conditions: (a)
ε ε
⇒
R2ε ⊂ R2ε ;
(b)
ε=0
⇒
R2ε = R2 .
(1)
The mapping f : X → Y , which satisfies the condition ∀x,y∈X
x R1 y
⇒
f (x) R2ε f (y)
(2)
is called ε-relation transferring. We say that a relation S ⊂ Y × Y is weakly homogeneous, if x S y implies αx S αy for all α ∈ K and for all x, y ∈ Y . A relation S ⊂ Y × Y is called homogeneous, if x S y implies αx S βy for arbitrary α, β ∈ K and all x, y ∈ Y . Next, we say that a family of relations R2ε ⊂ Y × Y for ε ∈ [0, 1) is continuous with respect to the relation R2 ⊂ Y × Y , if for each sequence (εn )n∈N such that 0 εn < 1, limn→+∞ εn = 0 and for all sequences (an )n∈N , (bn )n∈N ⊂ Y such that limn→+∞ an = a, limn→+∞ bn = b we have (∀n∈N an R2εn bn ) ⇒ a R2 b. For example, the relations ⊥, ⊥ε are weakly homogeneous and the family of relations ⊥ε for ε ∈ [0, 1) is continuous with respect to the relation ⊥. 3. General case Throughout this section, X, Y denote normed spaces (over the same field K). We are going to prove a stability of linear mappings transferring a relation. This result will be useful for further considerations. Lemma 1. Let R1 ⊂ X × X, R2ε ⊂ Y × Y be relations. Assume that R2ε is weakly homogeneous. If f : X → Y satisfies (2), then for α ∈ K the mapping αf also satisfies (2). Proof. Assume x R1 y. Then f (x) R2ε f (y). Since R2ε is weakly homogeneous, we obtain αf (x) R2ε αf (y), so αf satisfies (2). 2 Lemma 2. Let R1 ⊂ X × X, R2ε ⊂ Y × Y be relations. Suppose that a family {R2ε }ε∈[0,1) is continuous with respect to the relation R2 . Assume that limn→+∞ εn = 0 with εn ∈ [0, 1). If the mappings fn : X → Y are εn -relation transferring (for all n ∈ N) and
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satisfy limn→+∞ fn = f (pointwisely), then the mapping f : X → Y transfers relation (R1 into R2 ). Proof. Assume x R1 y. Then fn (x) R2εn fn (y). Since limn→+∞ fn = f and a family {R2ε }ε∈[0,1) is continuous with respect R2 , we get f (x) R2 f (y). Thus f preserves relation. 2 Now, we are able to prove the stability of the transferring relation property. The following theorem is the main theorem in this paper. Theorem 3. Let X, Y be finite-dimensional normed spaces and let R1 ⊂ X ×X, R2 , R2ε ⊂ Y × Y . Suppose that relations R2ε are weakly homogeneous for all ε ∈ [0, 1) and let the family of relations {R2ε }ε∈[0,1) be continuous with respect to the relation R2 . Then, for an arbitrary δ > 0, there exists ε > 0 such that for any linear ε-relation transferring mapping f : X → Y , there exists a linear mapping g: X → Y which transfers relation, such that f − g δ min f , g .
(3)
Proof. The case f = 0 is trivial. Then g := 0. Thus we may consider only nonzero linear mappings. First, we observe that (3) holds if and only if f − g δ f and f − g δ g
(4)
hold. Suppose that the assertion is not true, that there exists δ > 0 such that for each εn := n1 there exists a linear mapping εn -relation transferring fn : X → Y such that for each linear mapping g: X → Y which transfers relation, (3) is not true. Therefore from (4) we get fn − g > δ fn
or fn − g > δ g ,
whence fn g − fn fn > δ
fn g or − g g > δ.
(5)
Note that ffnn ∈ B(0, 1) ⊂ L(X; Y ) for all n ∈ N (here B(0, 1) denotes a closed unit ball). Since L(X; Y ) is finite-dimensional, the closed unit ball is compact. Therefore there f exists f ∈ L(X; Y ) and a subsequence ( fnnk )k∈N that k
lim
k→+∞
fnk = f. fnk
(6)
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f
By Lemma 1, for all k ∈ N, the mappings fnnk are εnk -relation transferring. Lemma 2 k yields that f exactly transfers the relation, and from (6) we have f = 1. Fix k ∈ N. Now we are able to substitute g by fnk · f in (5) fnk fnk fnk · f fnk · f > δ or − − fn · f fn · f > δ. fn fnk k k k Since f = 1, we have, for an arbitrary fixed k: fnk fn − f > δ, k which contradicts (6). 2 4. Orthogonality preserving property and stability problem For a normed space X (over the field K ∈ {R, C}) we consider ⊥i , ⊥εi and ⊥B , ⊥εB . It is easy to check that the both of families of relations {ε⊥i }ε∈[0,1) , {ε⊥B }ε∈[0,1) satisfy conditions (1). Moreover, ⊥i , ⊥B , ε⊥i , ε⊥B are weakly homogeneous. We say that f : X → Y preserves the i-orthogonality (or preserves the Birkhoff orthogonality), if it satisfies ∀x,y∈X
x ⊥i y
⇒
f (x) ⊥i f (y),
or
∀x,y∈X x ⊥B y ⇒ f (x) ⊥B f (y) .
Clearly f transfers relation ⊥i ⊂ X × X into ⊥i ⊂ Y × Y (or ⊥B ⊂ X × X into ⊥B ⊂ Y × Y ). Koldobsky, Blanco and Turnšek (cf. [8,1]) proved the following result, which describes the class of linear mappings preserving B-orthogonality. Theorem 4. Let f : X → Y be a linear operator. Then f preserves the Birkhoff–James orthogonality, if and only if, for some γ > 0, f (x) = γ x , x ∈ X. Now, we can define mappings, which approximately preserve the i-orthogonality (or approximately preserve the Birkhoff orthogonality). For ε ∈ [0, 1), f : X → Y is called an ε-i-orthogonality preserving mapping (or an ε-B-orthogonality preserving mapping), if it satisfies ∀x,y∈X
x ⊥i y
⇒
f (x) ε⊥i f (y),
(7)
∀x,y∈X x ⊥B y ⇒ f (x) ε⊥B f (y) .
(8)
or
Then, we may to say that f is ε-relation transferring.
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The following characterization of i-orthogonality preserving mappings was obtained in [6]. Theorem 5. Let f : X → Y be a linear operator. Then f is an ε-i-orthogonality preserving mapping, if and only if, ( 1−ε 1+ε ) f · x f x f · x , x ∈ X. Now, we obtain the results which are connected with the stability problem. Theorem 6. Let X, Y be finite-dimensional normed spaces. Then, for an arbitrary δ > 0 there exists ε > 0 such that for any linear mapping f : X → Y satisfying (7) there exists a linear mapping g: X → Y preserving i-orthogonality such that f − g δ · min f , g . Proof. Clearly relation ε⊥i is weakly homogeneous and the family {ε⊥i }ε∈[0,1) is continuous with respect to the relation ⊥i . Moreover, family {ε⊥i }ε∈[0,1) satisfies (1). Since f satisfies (7), so we can say that f is ε-relation transferring. Thus we are able to apply Theorem 3 and we obtain assertion. 2 In the same manner we can obtain the following theorem. Theorem 7. Let X, Y be finite-dimensional normed spaces. Then, for an arbitrary δ > 0, there exists ε > 0 such that for any linear ε-B-orthogonality preserving mapping f : X → Y there exists a linear B-orthogonality preserving mapping g: X → Y such that f − g δ min f , g . Theorems 6, 7 were obtained in the other way (see [6] and [9, Proposition 4.4]). But here we gave new proof and next we will obtain results for others relations. Now, for normed spaces, we can consider others notions of orthogonality. One of them is the Roberts orthogonality (R-orthogonality): x ⊥R y
:⇔
∀λ∈R
x + λy = x − λy ,
and an approximate Roberts orthogonality: x ⊥εR y
:⇔
∀λ∈R
x + λy 2 − x − λy 2 4ε x · λy .
If the norm comes from the an inner product, then ⊥ = ⊥R and ⊥ε = ⊥εR . Next, we could consider the class of linear mappings preserving R-orthogonality: ∀x,y∈X
x ⊥R y
⇒
f (x) ⊥R f (y),
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and ∀x,y∈X
x ⊥R y
⇒
f (x) ⊥εR f (y).
Similarly, for the Pythagorean orthogonality (P-orthogonality): x ⊥P y
:⇔
x + y 2 = x 2 + y 2 ,
and an approximate P-orthogonality: x ⊥εP y
:⇔
x + y 2 − x 2 − y 2 2ε x · y ,
one can consider mappings preserving P-orthogonality and their stability. Of course, in an inner product space we have ⊥ = ⊥P and ⊥ε = ⊥εP . Applying Theorem 3 we get that for R-orthogonality and P-orthogonality the stability problem has an affirmative answer for finite-dimensional normed spaces. The case of infinitely-dimensional spaces remains an open problem. 5. Linear mappings which preserve angle and stability problem For real inner product space (X, ·|·) we define the relation which is connected with the notion of angle. Fix c ∈ (−1, 1). For x, y ∈ X we define x ∠c y
:⇔
x|y = c x · y .
It means that the size of the angle between the vectors x and y equals arccos(c). We say that f : X → Y preserves the angle, if it satisfies ∀x,y∈X
x ∠c y
⇒
f (x) ∠c f (y).
(9)
It is easy to see that ∠0 = ⊥. Next, we fix ε ∈ [0, 1) and we define x ∠εc y
:⇔
x|y − c x · y ε x · y .
It means that the size of the angle satisfies c − ε arccos(c) c + ε. The linear mapping f : X → Y which satisfies the condition ∀x,y∈X
x ∠c y
⇒
f (x) ∠εc f (y)
(10)
is called approximately angle preserving. It is easy to check that the family of relations {∠εc }ε∈[0,1) satisfies conditions (1). Moreover, ∠c , ∠εc are weakly homogeneous. Therefore, using Theorem 3, we obtain the following theorem.
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Theorem 8. Let X, Y be finite-dimensional inner product spaces. Then for an arbitrary δ > 0 there exists ε > 0 such that for any linear mapping f : X → Y satisfying (10) there exists a linear mapping g: X → Y satisfying (9) such that f − g δ · min f , g . 6. Counterexample We now turn our attention to showing that for B-orthogonality the stability problem has a negative answer for infinite-dimensional normed spaces. The examples from [6] and from [9] concern the case of an infinite-dimensional domain X and an infinite-dimensional range Y . Therefore the following question appears. Does the problem of stability have a positive answer in the case of finite-dimensional domain X and infinite-dimensional range Y ? Let X, Y be normed spaces. A linear mapping f : X → Y is called an ε-isometry if it satisfies ∀x∈X
(1 − ε) x f (x) (1 + ε) x .
(11)
The following lemma will be useful. Lemma 9. Every linear ε-isometry is an ε -B-orthogonality preserving mapping, with ε = 1 − 1−ε 1+ε . Proof. Let x, y ∈ X and assume that x ⊥B y, i.e., x x + αy for all α ∈ K. Let λ ∈ K. Then from (11) we have f (x) (1 + ε) x (1 + ε) x + λy (1 + ε) 1 f (x + λy) 1−ε 1 + ε f (x) + λf (y), = 1−ε hence for any λ ∈ K we get (1 − ε ) f (x) f (x) + λf (y) and f is ε -B-orthogonality preserving. 2 Example 10. Next, we consider the space R2 with the standard inner products. We denote by H the space R2 with the Euclidean norm · H coming from this inner product. We can inscribe a regular polygon which has 2n sides inside a closed unit ball B((0, 0); 1) ⊂ H. Let Pn denote this “2n -regular polygon”. The set Pn is convex, absorbed and balanced. Hence this set introduces a new norm · n by the Minkowski functional. It is easy to see that Pn ⊂ B((0, 0); 1) and αn B((0, 0); 1) ⊂ Pn for some αn ∈ (0, 1). Obviously, αn 1, if n → +∞. So, we get αn x n x H x n . Let Xn denote the space R2 with the norm · n .
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Next, we define a space Y := y = (y1 , y2 , y3 , . . . , yn , 0, 0, . . .) : yk ∈ R2 , n ∈ N and we introduce a norm y Y := max y1 H , y2 2 , y3 3 , . . . , yn n for y = (y1 , y2 , . . . , yn , 0, 0, . . .) ∈ Y . Clearly dim H = 2, dim Y = ∞. Now, we define a linear mapping fk : H → Y by fk (x) := (0, . . . , 0, x, 0, 0, . . .), where x appears only at the kth coordinate. We will prove that fk is an εk -isometry with εk := α1k − 1 (hence εk 0). For any x ∈ H, it follows that (1 − εk ) x H x H x k
1 x H = (1 + εk ) x H . αk
Since fk (x) Y = x k we have ∀x∈X
(1 − εk ) x H fk (x)Y (1 + εk ) x H .
Let W be any normed space. By S(W ), B(W ) we denote a unit sphere and a closed unit ball, respectively. The following lemma gives us a characterization of linear isometry h: H → Y . Note that for some n ∈ N we have h(x) = h1 (x), h2 (x), h3 (x), . . . , hn (x), 0, 0, . . . , where each of the mappings h1 : H → H, hm : H → Xm (for m = 2, 3, . . . , n) is linear (the number n depends only on h). Lemma 11. If h: H → Y is linear isometry, then h1 = 1. Proof. Suppose that h1 < 1. Then there is c ∈ (0, 1) that for all x ∈ S(H) we have h1 (x) H c < 1. We define
h: H → Y by
h(x) := (0, h2 (x), h3 (x), . . . , hn (x), 0, 0, . . .). For any x ∈ S(H) we get 1 = h(x) = max h1 (x)H , h2 (x)2 , h3 (x)3 , . . . , hn (x)n = max h2 (x)2 , h3 (x)3 , . . . , hn (x)n =
h(x), and thus the mapping
h is also isometry. Now, we might write
h: H → X2 × . . . × Xn . A routine application of mathematical induction shows that the Cartesian product of finite number of simplexes is a simplex. Since a norm in X2 × . . . × Xn is given by u = max{ u2 2 , u3 3 , . . . , un n }, then it can be shown that
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B(X2 × . . . × Xn ) = B(X2 ) × . . . × B(Xn ). Each of the sets B(X2 ), . . . , B(Xn ) is a simplex. Therefore the closed unit ball in these normed spaces X2 × . . . × Xn is also a simplex. It is easy to see that
h(H) is an isometry and a set
h: H →
h(H) ∩ B(X2 × . . . × Xn )
is a closed unit ball in the space h(H). It is well-known that the intersection of twodimensional space and simplex is a simplex. Therefore the closed unit ball
h(H) ∩B(X2 × . . . × Xn ) is a simplex in the space
h(H). Thus the space
h(H) is not strictly convex. On the other hand, the space H is strictly convex, which contradicts the isometry
h(H). 2 h: H →
Further, we show that for B-orthogonality the stability problem has a negative answer. By Lemma 9, the mappings fk : H → Y are ε k -B-orthogonality preserving mapping, 1 k with ε k = 1 − 1−ε k → 0). 1+εk , where εk = αn − 1 (and ε Suppose that the problem of stability has a positive answer. Let 0 < δ < 1. Then, for ε k sufficiently small (i.e., k sufficiently large), there exists some linear mapping gk : H → Y preserving B-orthogonality such that fk − gk δ min fk , gk .
(12)
The mapping gk is a linear similarity by Theorem 4. It is easy to see that for some linear isometry hk : H → Y there is gk = gk hk . From (12) we get fk − gk δ gk .
(13)
Let h = (h1 , h2 , . . . , hn , 0, 0, . . .): H → Y be any isometry. Then there is xo ∈ S(H) such that h1 (xo ) H = 1 by Lemma 11. For all γ > 0, we obtain fk − γh fk (xo ) − γh(xo )Y = (0, . . . , 0, xo , 0, 0, . . .) − γ h1 (xo ), h2 (xo ), . . . , hn (xo ), 0, 0, . . . = max γ h1 (xo )H , . . . γ h1 (xo )H = γ. For the arbitrary mapping gk , there is an isometry hk such that gk = gk hk . If we put gk instead of γ and hk instead of h in the previous inequality, then the above consideration yields gk fk − gk . This fact is conflicting with (13). References [1] A. Blanco, A. Turnšek, On maps that preserve orthogonality in normed spaces, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 709–716. [2] J. Chmieliński, On an ε-Birkhoff orthogonality, JIPAM. J. Inequal. Pure Appl. Math. 6 (3) (2005), Art. 79. [3] J. Chmieliński, Remarks on orthogonality preserving mappings in normed spaces and some stability problems, Banach J. Math. Anal. 1 (1) (2007) 117–124.
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[4] J. Chmieliński, Linear mappings approximately preserving orthogonality, J. Math. Anal. Appl. 304 (2005) 158–169. [5] J. Chmieliński, Stability of the orthogonality preserving property in finite-dimensional inner product spaces, J. Math. Anal. Appl. 318 (2006) 433–443. [6] J. Chmieliński, P. Wójcik, Isosceles-orthogonality preserving property and its stability, Nonlinear Anal. 72 (2010) 1445–1453. [7] S.S. Dragomir, On approximation of continuous linear functionals in normed linear spaces, An. Univ. Timiş. Ser. Ştiinţ. Mat. 29 (1991) 51–58. [8] A. Koldobsky, Operators preserving orthogonality are isometries, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 835–837. [9] B. Mojškerc, A. Turnšek, Mappings approximately preserving orthogonality in normed spaces, Nonlinear Anal. 73 (2010) 3821–3831. [10] A. Turnšek, On mappings approximately preserving orthogonality, J. Math. Anal. Appl. 336 (2007) 625–631.