Journal of Inequalities in Pure and
Applied Mathematics
ON SOME APPROXIMATE FUNCTIONAL RELATIONS STEMMING
FROM ORTHOGONALITY PRESERVING PROPERTY
volume 7, issue 3, article 85,
2006.
JACEK CHMIELIŃSKI
Instytut Matematyki, Akademia Pedagogiczna w Krakowie
Podchora̧żych 2,
30-084 Kraków, Poland
Received 17 January, 2006;
accepted 24 February, 2006.
Communicated by: K. Nikodem
EMail: jacek@ap.krakow.pl
Abstract
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c
2000
Victoria University
ISSN (electronic): 1443-5756
015-06
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Abstract
Referring to previous papers on orthogonality preserving mappings we deal
with some relations, connected with orthogonality, which are preserved exactly
or approximately. In particular, we investigate the class of mappings approximately preserving the right-angle. We show some properties similar to those
characterizing mappings which exactly preserve the right-angle. Besides, some
kind of stability of the considered property is established. We study also the
property that a particular value c of the inner product is preserved. We compare the case c 6= 0 with c = 0, i.e., with orthogonality preserving property. Also
here some stability results are given.
On Some Approximate
Functional Relations Stemming
from Orthogonality Preserving
Property
Jacek Chmieliński
2000 Mathematics Subject Classification: 39B52, 39B82, 47H14.
Key words: Orthogonality preserving mappings, Right-angle preserving mappings,
Preservation of the inner product, Stability of functional equations.
Contents
1
2
Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Additivity of Approximately Right-angle Preserving Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3
Approximate Right-angle Preserving Mappings are Approximate Similarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4
Some Stability Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5
On Mappings Preserving a Particular Value of the Inner
Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References
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1.
Prerequisites
Let X and Y be real inner product spaces with the standard orthogonality relation ⊥. For a mapping f : X → Y it is natural to consider the orthogonality
preserving property:
∀x, y ∈ X : x⊥y ⇒ f (x)⊥f (y).
(OP)
The class of solutions of (OP) contains also very irregular mappings (cf. [1,
Examples 1 and 2]). On the other hand, a linear solution f of (OP) has to be a
linear similarity, i.e., it satisfies (cf. [1, Theorem 1])
kf (x)k = γkxk,
(1.1)
x∈X
On Some Approximate
Functional Relations Stemming
from Orthogonality Preserving
Property
Jacek Chmieliński
or, equivalently,
(1.2)
hf (x)|f (y)i = γ 2 hx|yi ,
x, y ∈ X
with some γ ≥ 0 (γ > 0 for f 6= 0). (More generally, a linear mapping between
real normed spaces which preserves the Birkhoff-James orthogonality has to
satisfy (1.1) – see [5].) Therefore, linear orthogonality preserving mappings are
not far from inner product preserving mappings (linear isometries), i.e., solutions of the functional equation:
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∀x, y ∈ X : hf (x)|f (y)i = hx|yi .
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A property similar to (OP) was introduced by Kestelman and Tissier (see
[6]). One says that f has the right-angle preserving property iff:
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(1.3)
(RAP)
∀x, y, z ∈ X : x − z⊥y − z ⇒ f (x) − f (z)⊥f (y) − f (z).
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For the solutions of (RAP) it is known (see [6]) that they must be affine, continuous similarities (with respect to some point).
It is easily seen that if f satisfies (RAP) then, for an arbitrary y0 ∈ Y , the
mapping f + y0 satisfies (RAP) as well. In particular, f0 := f − f (0) satisfies
(RAP) and f0 (0) = 0.
Summing up we have:
Theorem 1.1. The following conditions are equivalent:
(i) f satisfies (RAP) and f (0) = 0;
(ii) f satisfies (OP) and f is continuous and linear;
(iii) f satisfies (OP) and f is linear;
(iv) f is linear and satisfies (1.1) for some constant γ ≥ 0;
(v) f satisfies (1.2) for some constant γ ≥ 0;
(vi) f satisfies (OP) and f is additive.
Proof. (i)⇒(ii) follows from [6] (see above); (ii)⇒(iii) is trivial; (iii)⇒(iv) follows from [1] (see above); (iv)⇒(v) by use of the polarization formula and
(v)⇒(vi)⇒(i) is trivial.
In particular, one can consider a real vector space X with two inner products h·|·i1 and h·|·i2 and f = id|X a linear and continuous mapping between
(X, h·|·i1 ) and (X, h·|·i2 ). Then we obtain from Theorem 1.1:
On Some Approximate
Functional Relations Stemming
from Orthogonality Preserving
Property
Jacek Chmieliński
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Corollary 1.2. Let X be a real vector space equipped with two inner products
h·|·i1 and h·|·i2 generating the norms k · k1 , k · k2 and orthogonality relations
⊥1 , ⊥2 , respectively. Then the following conditions are equivalent:
(i) ∀x, y, z ∈ X : x − z⊥1 y − z ⇒ x − z⊥2 y − z;
(ii) ∀x, y ∈ X : x⊥1 y ⇒ x⊥2 y;
(iii) kxk2 = γkxk1 for x ∈ X with some constant γ > 0;
(iv) hx|yi2 = γ 2 hx|yi1 for x, y ∈ X with some constant γ > 0;
(v) ∀x, y, z ∈ X : x − z⊥1 y − z ⇔ x − z⊥2 y − z;
(vi) ∀x, y ∈ X : x⊥1 y ⇔ x⊥2 y.
For ε ∈ [0, 1) we define an ε-orthogonality by
ε
u⊥ v :⇔ | hu|vi | ≤ εkukkvk.
(Some remarks on how to extend this definition to normed or semi-inner product
spaces can be found in [2].)
Then, it is natural to consider an approximate orthogonality preserving (a.o.p.)
property:
(ε-OP)
∀x, y ∈ X : x⊥y ⇒ f (x)⊥ε f (y)
and the approximate right-angle preserving (a.r.a.p.) property:
(ε-RAP)
∀x, y, z ∈ X : x − z⊥y − z ⇒ f (x) − f (z)⊥ε f (y) − f (z).
On Some Approximate
Functional Relations Stemming
from Orthogonality Preserving
Property
Jacek Chmieliński
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The class of linear mappings satisfying (ε-OP) has been considered by the
author (cf. [1, 3]). In the present paper we are going to deal with mappings satisfying (ε-RAP) and, in the last section, with mappings which preserve (exactly
or approximately) a given value of the inner product. We will deal also with
some stability problems. (For basic facts concerning the background and main
results in the theory of stability of functional equations we refer to [4].) The
following result establishing the stability of equation (1.3) has been proved in
[3] and will be used later on.
Theorem 1.3 ([3], Theorem 2). Let X and Y be inner product spaces and let X
be finite-dimensional. Then, there exists a continuous mapping δ : R+ → R+
such that limε→0+ δ(ε) = 0 which satisfies the following property: For each
mapping f : X → Y (not necessarily linear) satisfying
(1.4)
| hf (x)|f (y)i − hx|yi | ≤ εkxkkyk,
x, y ∈ X
there exists a linear isometry I : X → Y such that
kf (x) − I(x)k ≤ δ(ε)kxk,
On Some Approximate
Functional Relations Stemming
from Orthogonality Preserving
Property
Jacek Chmieliński
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x ∈ X.
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2.
Additivity of Approximately Right-angle
Preserving Mappings
Tissier [6] showed that a mapping f satisfying the (RAP) property has to be additive up to a constant f (0). Following his idea we will show that a.r.a.p. mappings are, in some sense, quasi-additive. We start with the following lemma.
Lemma 2.1. Let X be a real inner product space. Let a set of points a, b, c, d, e ∈
X satisfies the following relations, with ε ∈ [0, 18 ),
(2.1)
a − b⊥ε c − b,
b − c⊥ε d − c,
c − d⊥ε a − d,
d − a⊥ε b − a;
On Some Approximate
Functional Relations Stemming
from Orthogonality Preserving
Property
Jacek Chmieliński
(2.2)
a − e⊥ε b − e,
b − e⊥ε c − e,
c − e⊥ε d − e,
d − e⊥ε a − e.
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Then,
with δ :=
a
+
c
≤ δka − ck
e −
2 q
3ε
.
1−4ε
Proof. We have
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kc − ek2 = ke − a + a − ck2 = ke − ak2 + ka − ck2 + 2 he − a|a − ci
whence
kc − ek2 − ka − ek2 − ka − ck2
he − a|a − ci =
.
2
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Thus
2 2
e − a + c = e − a + a − c 2
2 1
= ke − ak2 + ka − ck2 + he − a|a − ci
4
1
1
1
1
= ke − ak2 + ka − ck2 + kc − ek2 − ka − ek2 − ka − ck2
4
2
2
2
1
1
1
= ka − ek2 + kc − ek2 − ka − ck2 .
2
2
4
Finally,
(2.3)
2
a
+
c
= 1 2ka − ek2 + 2kc − ek2 − ka − ck2
e −
2 4
and, analogously,
2
b
+
d
e −
= 1 2kb − ek2 + 2kd − ek2 − kb − dk2 .
(2.4)
2
4
Adding the equalities:
ka − bk2
kb − ck2
kc − dk2
kd − ak2
= ka − e + e − bk2 = ka − ek2 + ke − bk2 + 2 ha − e|e − bi ,
= kb − e + e − ck2 = kb − ek2 + ke − ck2 + 2 hb − e|e − ci ,
= kc − e + e − dk2 = kc − ek2 + ke − dk2 + 2 hc − e|e − di ,
= kd − e + e − ak2 = kd − ek2 + ke − ak2 + 2 hd − e|e − ai
On Some Approximate
Functional Relations Stemming
from Orthogonality Preserving
Property
Jacek Chmieliński
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one gets
(2.5) ka − bk2 + kb − ck2 + kc − dk2 + kd − ak2
= 2ka − ek2 + 2kb − ek2 + 2kc − ek2 + 2kd − ek2
+ 2 ha − e|e − bi + 2 hb − e|e − ci
+ 2 hc − e|e − di + 2 hd − e|e − ai .
Similarly, adding
ka − ck2
ka − ck2
kb − dk2
kb − dk2
= ka − b + b − ck2 = ka − bk2 + kb − ck2 + 2 ha − b|b − ci ,
= ka − d + d − ck2 = ka − dk2 + kd − ck2 + 2 ha − d|d − ci ,
= kb − a + a − dk2 = kb − ak2 + ka − dk2 + 2 hb − a|a − di ,
= kb − c + c − dk2 = kb − ck2 + kc − dk2 + 2 hb − c|c − di
On Some Approximate
Functional Relations Stemming
from Orthogonality Preserving
Property
Jacek Chmieliński
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one gets
(2.6) ka − ck2 + kb − dk2
= ka − bk2 + ka − dk2 + kc − bk2 + kc − dk2
+ ha − b|b − ci + ha − d|d − ci
+ hb − a|a − di + hb − c|c − di .
Using (2.3) – (2.6) we derive
2 2
a
+
c
b
+
d
e −
+ e −
2 2 Contents
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(2.3),(2.4)
=
2ka − ek2 + 2kc − ek2 + 2kb − ek2 + 2kd − ek2 − ka − ck2 − kb − dk2
4
1
kb − ck2 + kc − dk2 + kd − ak2 + ka − bk2
4
− 2 hb − e|e − ci − 2 hc − e|e − di − 2 hd − e|e − ai
− 2 ha − e|e − bi − ka − ck2 − kb − dk2
1
(2.6)
= − ha − b|b − ci + ha − d|d − ci + hb − a|a − di + hb − c|c − di
4
+ 2 hb − e|e − ci + 2 hc − e|e − di + 2 hd − e|e − ai + 2 ha − e|e − bi .
(2.5)
=
Thus
2 2
e − a + c + e − b + d 2
2 1
≤
| ha − b|b − ci | + | ha − d|d − ci | + | hb − a|a − di |
4
+ | hb − c|c − di | + 2| hb − e|e − ci | + 2| hc − e|e − di |
+ 2| hd − e|e − ai | + 2| ha − e|e − bi | .
Using the assumptions (2.1) and (2.2), we obtain
2 2
a
+
c
b
+
d
+ e −
e −
(2.7)
2 2 1 ≤ ε ka − bkkb − ck + ka − dkkd − ck + kb − akka − dk
4
+ kb − ckkc − dk + 2kb − ekke − ck + 2kc − ekke − dk
On Some Approximate
Functional Relations Stemming
from Orthogonality Preserving
Property
Jacek Chmieliński
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+ 2kd − ekke − ak + 2ka − ekke − bk
1 = ε (kb − ck + ka − dk)(ka − bk + kc − dk)
4
+ 2(kb − ek + kd − ek)(ka − ek + kc − ek) .
Notice, that for ε = 0, (2.7) yields e =
Let
a+c
2
=
b+d
.
2
% := max{ka − bk, kb − ck, kc − dk, kd − ak, ka − ek, kb − ek, kc − ek, kd − ek}.
It follows from (2.7) that
2 2
e − a + c + e − b + d ≤ 1 ε(2% · 2% + 2 · 2% · 2%) = 3ε%2 .
2 2 4
Jacek Chmieliński
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Then, in particular
(2.8)
On Some Approximate
Functional Relations Stemming
from Orthogonality Preserving
Property
2
e − a + c ≤ 3ε%2 .
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Since we do not know for which distance the value % is attained, we are
going to consider a few cases.
1. % ∈ {ka − bk, kb − ck, kc − dk, kd − ak}.
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Suppose that % = ka−bk (other possibilities in this case are similar). Then
ka − ck2 = ka − b + b − ck2
= ka − bk2 + kb − ck2 + 2 ha − b|b − ci
≥ %2 + 0 − 2εka − bkkb − ck
≥ %2 − 2ε%2 = (1 − 2ε)%2 .
Assuming ε < 21 ,
%2 ≤
1
ka − ck2 .
1 − 2ε
On Some Approximate
Functional Relations Stemming
from Orthogonality Preserving
Property
Using (2.8) we have
Jacek Chmieliński
2
a
+
c
e −
≤ 3ε ka − ck2
2 1 − 2ε
whence
(2.9)
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r
a
+
c
3ε
e −
≤
ka − ck.
2
1 − 2ε
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2. % ∈ {ka − ek, kc − ek}.
Suppose that % = ka − ek (the other possibility
(2.3), we have
2
1
a
+
c
= 1 ka − ek2 +
ka − ck2 + e
−
4
2 2
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is similar). Then, from
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1
1
kc − ek2 ≥ %2 ,
2
2
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whence
2
1
a + c
2
.
% ≤ ka − ck + 2 e −
2
2 2
From it and (2.8) we get
2
2
a
+
c
3ε
a
+
c
2
2
e −
≤ 3ε% ≤ ka − ck + 6ε e −
2 2
2 whence (assuming ε < 16 )
(2.10)
s
3ε
a
+
c
e −
≤
ka − ck.
2(1 − 6ε)
2
3. % ∈ {kb − ek, kd − ek}.
On Some Approximate
Functional Relations Stemming
from Orthogonality Preserving
Property
Jacek Chmieliński
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Suppose that % = kb − ek (the other possibility is similar). We have then
kb − ak2 = kb − e + e − ak2
= kb − ek2 + ke − ak2 + 2 hb − e|e − ai
≥ %2 + 0 − 2εkb − ekke − ak
≥ %2 − 2ε%2 = (1 − 2ε)%2 ,
whence
kb − ak2 ≥ (1 − 2ε)%2 .
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Using this estimation we have
ka − ck2 = ka − b + b − ck2
= ka − bk2 + kb − ck2 + 2 ha − b|b − ci
≥ (1 − 2ε)%2 + 0 − 2εka − bkkb − ck
≥ (1 − 2ε)%2 − 2ε%2
= (1 − 4ε)%2 ,
whence (for ε < 41 )
%2 ≤
On Some Approximate
Functional Relations Stemming
from Orthogonality Preserving
Property
1
ka − ck2 .
1 − 4ε
Using (2.8) we get
2
a
+
c
≤ 3ε%2 ≤ 3ε ka − ck2
e −
2 1 − 4ε
Jacek Chmieliński
and
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r
a
+
c
3ε
e −
≤
ka − ck.
2
1 − 4ε
(2.11)
Finally, assuming ε <
(r
max
1
,
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we have
3ε
,
1 − 2ε
s
3ε
,
2(1 − 6ε)
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r
3ε
1 − 4ε
)
r
=
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3ε
1 − 4ε
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and it follows from (2.9) – (2.11)
r
a
+
c
3ε
≤
e −
ka − ck,
2
1 − 4ε
which completes the proof.
Theorem 2.2. Let X and Y be real inner product spaces and let f : X → Y
satisfy (ε-RAP) with ε < 18 . Then f satisfies
x+y
f (x) + f (y) ≤ δkf (x) − f (y)k for x, y ∈ X
(2.12) f
−
2
2
q
3ε
.
with δ = 1−4ε
for
Jacek Chmieliński
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Moreover, if additionally f (0) = 0, then f satisfies (ε-OP) and
(2.13) kf (x + y) − f (x) − f (y)k
≤ 2δ(kf (x + y)k + kf (x) − f (y)k),
On Some Approximate
Functional Relations Stemming
from Orthogonality Preserving
Property
x, y ∈ X.
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Proof. Fix arbitrarily x, y ∈ X. The case x = y is obvious. Assume x 6= y.
Choose u, v ∈ X such that x, u, y, v are consecutive vertices of a square with
the center at x+y
. Denote
2
x+y
a := f (x), b := f (u), c := f (y), d := f (v), e := f
.
2
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Since x−u⊥y −u, u−y⊥v −y, y −v⊥x−v, v −x⊥u−x and x− x+y
⊥u− x+y
,
2
2
x+y
x+y
x+y
x+y
x+y
x+y
u − 2 ⊥y − 2 , y − 2 ⊥v − 2 , v − 2 ⊥x − 2 , it follows from (ε-RAP)
that the conditions (2.1) and (2.2) are satisfied. The assertion of Lemma 2.1
yields (2.12).
For the second assertion, it is obvious that f satisfies (ε-OP). Inequality
(2.13) follows from (2.12). Indeed, putting y = 0 we get
x
f
(x)
f
≤ δkf (x)k,
−
x ∈ X.
2
2 Now, for x, y ∈ X
kf (x+y) − f (x) − f (y)k
f (x + y)
x
+
y
x
+
y
f
(x)
+
f
(y)
= 2
−
f
+
f
−
2
2
2
2
f (x + y)
x+y x+y
f (x) + f (y) ≤ 2
−f
−
+ 2 f
2
2
2
2
≤ 2δkf (x + y)k + 2δkf (x) − f (y)k.
On Some Approximate
Functional Relations Stemming
from Orthogonality Preserving
Property
Jacek Chmieliński
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For ε = 0 we obtain that if f satisfies (RAP) and f (0) = 0, then f is additive.
The following, reverse in a sense, statement is easily seen.
Lemma 2.3. If f satisfies (ε-OP) and f is additive, then f satisfies (ε-RAP) and
f (0) = 0.
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Example 2 in [1] shows that it is not possible to omit completely the additivity assumption in the above lemma. However, the problem arises if additivity
can be replaced by a weaker condition (e.g. by (2.13)). This problem remains
open.
On Some Approximate
Functional Relations Stemming
from Orthogonality Preserving
Property
Jacek Chmieliński
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3.
Approximate Right-angle Preserving Mappings
are Approximate Similarities
As we know from [6], a right-angle preserving mappings are similarities. Our
aim is to show that a.r.a.p. mappings behave similarly. We start with a technical
lemma.
Lemma 3.1. Let a, x ∈ X and ε ∈ [0, 1). Then
(a − x)⊥ε (−a − x)
(3.1)
if and only if
q
2ε
2
2
(3.2)
kxk − kak ≤ √
kak2 kxk2 − ha|xi2 .
2
1−ε
Moreover, it follows from (3.1) that
r
r
1−ε
1+ε
(3.3)
kak ≤ kxk ≤
kak.
1+ε
1−ε
Proof. The condition (3.1) is equivalent to:
| ha + x|a − xi | ≤ εka + xkka − xk,
p
p
kak2 − kxk2 ≤ ε kak2 + 2 ha|xi + kxk2 kak2 − 2 ha|xi + kxk2 ,
kak2 − kxk2
2
≤ ε2 (kak2 + kxk2 + 2 ha|xi)(kak2 + kxk2 − 2 ha|xi)
= ε2 (kak2 + kxk2 )2 − 4 ha|xi2
= ε2 (kak2 − kxk2 )2 + 4kak2 kxk2 − 4 ha|xi2 ,
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and finally
(1 − ε2 )(kak2 − kxk2 )2 ≤ 4ε2 kak2 kxk2 − ha|xi2
which is equivalent to (3.2).
Inequality (3.2) implies
kxk2 − kak2 ≤ √ 2ε kakkxk,
1 − ε2
which yields
kxk kak 2ε
kak − kxk ≤ √1 − ε2
(3.4)
(we assume x 6= 0 and a 6= 0, otherwise the assertion of the lemma is trivial).
2ε
Denoting t := kxk
> 0 and α := √1−ε
2 , the inequality (3.4) can be written in
kak
the form
|t − t−1 | ≤ α
with a solution
−α +
√
2
α2 + 4
≤t≤
α+
√
α2 + 4
.
2
On Some Approximate
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Therefore,
r
1−ε
kxk
≤
≤
1+ε
kak
r
1+ε
1−ε
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whence (3.3) is satisfied.
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Theorem 3.2. If f : X → Y is homogeneous and satisfies (ε-RAP), then with
some k ≥ 0:
3
3
1−ε 2
1+ε 2
(3.5)
k·
kxk ≤ kf (x)k ≤ k ·
kxk,
x ∈ X.
1+ε
1−ε
Proof. 1. For arbitrary x, y ∈ X we have
kxk = kyk ⇔ x − y⊥ − x − y.
It follows from (ε-RAP) and the oddness of f that
kxk = kyk ⇒ f (x) − f (y)⊥ε − f (x) − f (y).
On Some Approximate
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Lemma 3.1 yields
r
1−ε
kf (x)k ≤ kf (y)k ≤
1+ε
r
1+ε
kf (x)k.
1−ε
2. Fix arbitrarily x0 =
6 0 and define for r ≥ 0, ϕ(r) := f kxr0 k x0 . Using
(3.6) we have
r
r
1−ε
1+ε
kxk = r ⇒
ϕ(r) ≤ kf (x)k ≤
ϕ(r),
1+ε
1−ε
(3.6)
kxk = kyk ⇒
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whence
r
(3.7)
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1−ε
ϕ(kxk) ≤ kf (x)k ≤
1+ε
r
1+ε
ϕ(kxk),
1−ε
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x ∈ X.
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3. For t ≥ 0 and kxk = r we have
#
"r
r
1−ε
1+ε
kf (tx)k ∈
ϕ(tr),
ϕ(tr)
1+ε
1−ε
and
"r
tkf (x)k ∈
1−ε
tϕ(r),
1+ε
r
#
1+ε
tϕ(r) .
1−ε
Since kf (tx)k = tkf (x)k (homogeneity of f ),
#
# "r
"r
r
r
1+ε
1−ε
1+ε
1−ε
tϕ(r) 6= ∅.
tϕ(r),
ϕ(tr) ∩
ϕ(tr),
1−ε
1+ε
1−ε
1+ε
Thus there exist λ, µ ∈
hq
1−ε
,
1+ε
q
1+ε
1−ε
i
such that λϕ(tr) = µtϕ(r), whence
1−ε
1+ε
tϕ(r) ≤ ϕ(tr) ≤
tϕ(r).
1+ε
1−ε
In particular, for r = 1 and k := ϕ(1) we get
(3.8)
1−ε
1+ε
kt ≤ ϕ(t) ≤
kt,
1+ε
1−ε
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t ≥ 0.
4. Using (3.7) and (3.8), we get
r
r
1+ε
1+ε 1+ε
kf (x)k ≤
ϕ(kxk) ≤
·
kkxk
1−ε
1−ε 1−ε
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and
r
kf (x)k ≥
1−ε
ϕ(kxk) ≥
1+ε
r
1−ε 1−ε
·
kkxk
1+ε 1+ε
whence (3.5) is proved.
For ε = 0 we get kf (x)k = kkxk for x ∈ X and, from (2.13), f is additive
hence linear. Thus f is a similarity.
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4.
Some Stability Problems
The stability of the orthogonality preserving property has been studied in [3].
We present the main result from that paper which will be used in this section.
Theorem 4.1 ([3, Theorem 4]). Let X, Y be inner product spaces and let X
be finite-dimensional. Then, there exists a continuous function δ : [0, 1) →
[0, +∞) with the property limε→0+ δ(ε) = 0 such that for each linear mapping
f : X → Y satisfying (ε-OP) one finds a linear, orthogonality preserving one
T : X → Y such that
kf − T k ≤ δ(ε) min{kf k, kT k}.
The mapping δ depends, actually, only on the dimension of X. Immediately,
we have from the above theorem:
Corollary 4.2. Let X, Y be inner product spaces and let X be finite-dimensional.
Then, for each δ > 0 there exists ε > 0 such that for each linear mapping
f : X → Y satisfying ( ε-OP) one finds a linear, orthogonality preserving one
T : X → Y such that
kf − T k ≤ δ min{kf k, kT k}.
We start our considerations with the following observation.
Proposition 4.3. Let f : X → Y satisfy (RAP) and f (0) = 0. Suppose that
g : X → Y satisfies
kf (x) − g(x)k ≤ M kf (x)k,
x∈X
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(M +2)
with some constant M < 14 . Then g satisfies (ε-OP) with ε := M(1−M
and
)2
√
(4.1) kg(x + y) − g(x) − g(y)k ≤ 2 ε(kg(x)k + kg(y)k),
x, y ∈ X;
√
(4.2)
kg(λx) − λg(x)k ≤ 2 ε|λ|kg(x)k,
x ∈ X, λ ∈ R.
Proof. It follows from Theorem 1.1 that for some γ ≥ 0, f satisfies (1.2) and
(1.1). Thus we have for arbitrary x, y ∈ X:
| hg(x)|g(y)i − γ 2 hx|yi |
= | hg(x) − f (x)|g(y) − f (y)i + hg(x) − f (x)|f (y)i
+ hf (x)|g(y) − f (y)i |
≤ kg(x) − f (x)kkg(y) − f (y)k + kg(x) − f (x)kkf (y)k
+ kf (x)kkg(y) − f (y)k
2
≤ M kf (x)kkf (y)k + M kf (x)kkf (y)k + M kf (x)kkf (y)k
= M (M + 2)γ 2 kxkkyk.
Using [1, Lemma 2], we get, for arbitrary x, y ∈ X, λ ∈ R:
p
kg(x + y) − g(x) − g(y)k ≤ 2 M (M + 2)γ(kxk + kyk)
p
= 2 M (M + 2)(kf (x)k + kf (y)k);
p
kg(λx) − λg(x)k ≤ 2 M (M + 2)γ|λ|kxk
p
= 2 M (M + 2)|λ|kf (x)k.
Since kf (x)k − kg(x)k ≤ M kf (x)k,
kg(x)k
kf (x)k ≤
.
1−M
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Therefore
| hg(x)|g(y)i − γ 2 hx|yi | ≤
Putting ε :=
M (M + 2)
kg(x)kkg(y)k.
(1 − M )2
M (M +2)
(1−M )2
we have x⊥y ⇒ g(x)⊥ε g(y) and
√
kg(x + y) − g(x) − g(y)k ≤ 2 ε(kg(x)k + kg(y)k)
√
kg(λx) − λg(x)k ≤ 2 ε|λ|kg(x)k.
Corollary 4.4. If f satisfies (RAP), f (0) = 0 (whence f is linear) and g : X →
Y is a linear mapping satisfying, with M < 14 ,
(4.3)
On Some Approximate
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kf − gk ≤ M kf k,
then g is (ε-OP) (and linear) whence (ε-RAP).
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The above result yields a natural question if the reverse statement is true.
Namely, we may ask if for a linear mapping g : X → Y satisfying (ε-RAP)
(with some ε > 0) there exists a (linear) mapping f satisfying (RAP) such that
an estimation of the (4.3) type holds.
A particular solution to this problem follows easily from Theorem 4.1.
Contents
Theorem 4.5. Let X be a finite-dimensional inner product space and Y an arbitrary one. There exists a mapping δ : [0, 1) → R+ satisfying limε→0+ δ(ε) = 0
and such that for each linear mapping satisfying (ε-RAP) g : X → Y there
exists f : X → Y satisfying (RAP) and such that
(4.4)
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kf − gk ≤ δ(ε) min{kf k, kgk}.
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Proof. If g is linear and satisfies (ε-RAP), then g is linear and satisfies (ε-OP).
It follows then from Theorem 4.1 that there exists f linear and satisfying (OP),
whence (RAP), such that (4.4) holds.
Corollary 4.6. Let X be a finite-dimensional inner product space and Y an
arbitrary one. Then, for each δ > 0 there exists ε > 0 such that for each linear
and satisfying (ε-RAP) mapping g : X → Y there exists f : X → Y satisfying
(RAP) and such that
kf − gk ≤ δ min{kf k, kgk}.
It is an open problem to verify if the above result remains true in the infinite
dimensional case or without the linearity assumption.
On Some Approximate
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5.
On Mappings Preserving a Particular Value of
the Inner Product
The following considerations have been inspired by a question of L. Reich during the 43rd ISFE. In this section X and Y are inner product spaces over the
field K of real or complex numbers. Let f : X → Y and suppose that, for a
fixed number c ∈ K, f preserves this particular value of the inner product, i.e.,
(5.1)
∀x, y ∈ X : hx|yi = c ⇒ hf (x)|f (y)i = c.
If c = 0, the condition (5.1) simply means that f preserves orthogonality,
i.e., that f satisfies (OP).
We will show that the solutions of (5.1) behave differently for c = 0 and for
c 6= 0.
Obviously, if f satisfies (1.3) then f also satisfies (5.1), with an arbitrary c.
The converse is not true, neither with c = 0 (cf. examples mentioned in the
Introduction) nor with c 6= 0. Indeed, if c > 0, then fixing x0 ∈ X such that
kx0 k2 = c, a constant mapping f (x) = x0 , x ∈ X satisfies (5.1) but not (1.3).
Another example: let X = Y = C and let 0 6= c ∈ C. Define
c
f (z) := ,
z
z ∈ C \ {0};
f (0) := 0.
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Then, for z, w ∈ C \ {0}
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hf (z)|f (w)i =
|c|2
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hz|wi
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and, in particular, if hz|wi = c, then hf (z)|f (w)i = c. Thus f satisfies (5.1) but
not (1.3).
Let us restrict our investigations to the class of linear mappings. As we will
see below (Corollary 5.2), a linear solution of (5.1), with c 6= 0, satisfies (1.3).
Let us discuss a stability problem. For fixed 0 6= c ∈ K and ε ≥ 0 we
consider the condition
(5.2)
∀x, y ∈ X : hx|yi = c ⇒ | hf (x)|f (y)i − c| ≤ ε.
Theorem 5.1. For a finite-dimensional inner product space X and an arbitrary
inner product space Y there exists a continuous mapping δ : R+ → R+ satisfying limε→0+ δ(ε) = 0 and such that for each linear mapping f : X → Y
satisfying (5.2) there exists a linear isometry I : X → Y such that
kf − Ik ≤ δ(ε).
Proof. Let 0 6= d ∈ K. If hx|yi = d, then dc x|y = c and hence, using (5.2)
|c|
and homogeneity of f , |d|
| hf (x)|f (y)i − d| ≤ ε. Therefore we have (for d 6= 0)
(5.3)
hx|yi = d ⇒ | hf (x)|f (y)i − d| ≤
|d|
ε.
|c|
Now, let d = 0. Let 0 D
6= dn ∈ K and
E limn→∞ dn = 0. Suppose that hx|yi =
dn y
d = 0 and y 6= 0. Then x + kyk2 |y = dn and thus, from (5.3) and linearity of
f,
|dn |
d
n
f (x) +
≤
f
(y)|f
(y)
−
d
ε.
n
kyk2
|c|
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Letting n → ∞ we obtain hf (x)|f (y)i = 0. For y = 0 the latter equality is
obvious.
Summing up, we obtain that
| hf (x)|f (y)i − hx|yi | ≤
| hx|yi |
ε
|c|
whence also
(5.4)
| hf (x)|f (y)i − hx|yi | ≤
ε
kxkkyk.
|c|
Let δ 0 : R+ → R+ be a mapping from the assertion of Theorem 1.3. Define
δ(ε) := δ 0 (ε/|c|) and notice that limε→0+ δ(ε) = 0. Then it follows from (5.4)
that there exists a linear isometry I : X → Y such that
ε
0
kf − Ik ≤ δ
= δ(ε).
|c|
For ε = 0 we obtain from the above result (we can omit the assumption
concerning the dimension of X in this case, considering a subspace spanned on
given vectors x, y ∈ X):
Corollary 5.2. Let f : X → Y be linear and satisfy (5.1), with some 0 6= c ∈
K. Then f satisfies (1.3).
On Some Approximate
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Notice that for c = 0, the condition (5.2) has the form
hx|yi = 0 ⇒ | hf (x)|f (y)i | ≤ ε.
If hx|yi = 0, then also hnx|yi = 0 for all n ∈ N. Thus | hf (nx)|f (y)i | ≤ ε,
whence | hf (x)|f (y)i | ≤ nε for n ∈ N. Letting n → ∞ one gets
∀x, y ∈ X : hx|yi = 0 ⇒ hf (x)|f (y)i = 0,
i.e, f is a linear, orthogonality preserving mapping.
Now, let us replace the condition (5.2) by
(5.5)
∀x, y ∈ X : hx|yi = c ⇒ | hf (x)|f (y)i − c| ≤ εkf (x)kkf (y)k.
For c = 0, (5.5) states that f satisfies (ε-OP).
Let us consider the class of linear mappings satisfying (5.5) with c 6= 0.
We proceed similarly
as in the proof of Theorem 5.1. Let 0 6= d ∈ K. If
hx|yi = d, then dc x|y = c and hence, using (5.5) and homogeneity of f , we
obtain
|c|
|c|
| hf (x)|f (y)i − d| ≤ ε kf (x)kkf (y)k.
|d|
|d|
Therefore we have (for d 6= 0)
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hx|yi = d ⇒ | hf (x)|f (y)i − d| ≤ εkf (x)kkf (y)k.
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Now,
suppose
D
E that hx|yi = d = 0. Let 0 6= dn ∈ K and limn→∞ dn = 0. Then
dn y
x + kyk2 |y = dn and thus, from (5.6) and linearity of f ,
d
d
n
n
≤ ε f (x) +
kf (y)k.
f (x) +
f
(y)|f
(y)
−
d
f
(y)
n
kyk2
kyk2
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Letting n → ∞ we obtain | hf (x)|f (y)i | ≤ εkf (x)kkf (y)k. So we have
hx|yi = 0 ⇒ | hf (x)|f (y)i | ≤ εkf (x)kkf (y)k.
Summing up, we obtain that
(5.7)
| hf (x)|f (y)i − hx|yi | ≤ εkf (x)kkf (y)k,
Putting in the above inequality x = y we get kf (x)k ≤
gives
ε
| hf (x)|f (y)i − hx|yi | ≤
kxkkyk,
1−ε
ε
i.e., f satisfies (1.4) with the constant 1−ε
.
Therefore, applying Theorem 1.3, we get
x, y ∈ X.
kxk
√
1−ε
for x ∈ X, which
x, y ∈ X,
Theorem 5.3. If X is a finite-dimensional inner product space and Y an arbitrary inner product space, then there exists a continuous mapping δ : R+ → R+
with limε→0+ δ(ε) = 0 such that for a linear mapping f : X → Y satisfying
(5.5), with c 6= 0, there exists a linear isometry I : X → Y such that
kf − Ik ≤ δ(ε).
On Some Approximate
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Jacek Chmieliński
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A converse theorem is also true, even with no restrictions concerning the
dimension of X. Let I : X → Y be a linear isometry and f : X → Y a
mapping, not necessarily linear, such that
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kf (x) − I(x)k ≤ δkxk,
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q
with δ = 1+2ε
− 1 (for a given ε ≥ 0). Reasoning similarly as in the proof of
1+ε
Proposition 4.3 one can show that
| hf (x)|f (y)i − hx|yi | ≤ δ(δ + 2)kxkkyk,
which implies
1
kxk ≤ p
1 − δ(δ + 2)
kf (x)k
and finally
δ(δ + 2)
kf (x)kkf (y)k
1 − δ(δ + 2)
= εkf (x)kkf (y)k.
|hf (x)|f (y)i − hx|yi| ≤
Thus f satisfies (5.5) with an arbitrary c.
On Some Approximate
Functional Relations Stemming
from Orthogonality Preserving
Property
Jacek Chmieliński
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Remark 1. From Theorems 5.1 and 5.3 one can derive immediately the stability
results formulated as in Corollaries 4.2 and 4.6.
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References
[1] J. CHMIELIŃSKI, Linear mappings approximately preserving orthogonality, J. Math. Anal. Appl., 304 (2005), 158–169.
[2] J. CHMIELIŃSKI, On an ε-Birkhoff orthogonality, J. Inequal. Pure Appl.
Math., 6(3) (2005), Art. 79, 7pp. [ONLINE: http://jipam.vu.edu.
au/article.php?sid=552].
[3] J. CHMIELIŃSKI, Stability of the orthogonality preserving property in
finite-dimensional inner product spaces, J. Math. Anal. Appl., 318 (2006),
433–443.
[4] D.H. HYERS, G. ISAC AND Th.M. RASSIAS, Stability of Functional
Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications vol. 34, Birkhäuser, Boston-Basel-Berlin,
1998.
On Some Approximate
Functional Relations Stemming
from Orthogonality Preserving
Property
Jacek Chmieliński
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[5] A. KOLDOBSKY, Operators preserving orthogonality are isometries, Proc.
Roy. Soc. Edinburgh Sect. A, 123 (1993), 835–837.
[6] A. TISSIER, A right-angle preserving mapping (a solution of a problem
proposed in 1983 by H. Kestelman), Advanced Problem 6436, Amer. Math.
Monthly, 92 (1985), 291–292.
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