Tangent spaces to metric spaces and to their subspaces
We investigate tangent spaces and metric space valued derivatives at points of a general metric spaces. The conditions under which two different subspaces of the metric space have isometric tangent spaces in the common point of these subspaces are determinated.
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irk-123456789-109472010-08-11T12:02:05Z Tangent spaces to metric spaces and to their subspaces Dovgoshey, O.A. We investigate tangent spaces and metric space valued derivatives at points of a general metric spaces. The conditions under which two different subspaces of the metric space have isometric tangent spaces in the common point of these subspaces are determinated. 2008 Article Tangent spaces to metric spaces and to their subspaces / O.A. Dovgoshey // Український математичний вісник. — 2008. — Т. 5, № 4. — С. 470-487. — Бібліогр.: 12 назв. — англ. 1810-3200 http://dspace.nbuv.gov.ua/handle/123456789/10947 en Інститут прикладної математики і механіки НАН України |
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We investigate tangent spaces and metric space valued derivatives at points of a general metric spaces. The conditions under which two different subspaces of the metric space have isometric tangent spaces in the common point of these subspaces are determinated. |
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Dovgoshey, O.A. Tangent spaces to metric spaces and to their subspaces |
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Dovgoshey, O.A. |
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Dovgoshey, O.A. |
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Tangent spaces to metric spaces and to their subspaces |
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Tangent spaces to metric spaces and to their subspaces |
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Tangent spaces to metric spaces and to their subspaces |
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Tangent spaces to metric spaces and to their subspaces |
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Tangent spaces to metric spaces and to their subspaces |
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tangent spaces to metric spaces and to their subspaces |
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Інститут прикладної математики і механіки НАН України |
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2008 |
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Tangent spaces to metric spaces and to their subspaces / O.A. Dovgoshey // Український математичний вісник. — 2008. — Т. 5, № 4. — С. 470-487. — Бібліогр.: 12 назв. — англ. |
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AT dovgosheyoa tangentspacestometricspacesandtotheirsubspaces |
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2025-07-02T13:15:44Z |
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2025-07-02T13:15:44Z |
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1836541169807392768 |
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Український математичний вiсник
Том 5 (2008), № 4, 470 – 487
Tangent spaces to metric spaces
and to their subspaces
Oleksiy A. Dovgoshey
(Presented by V. Ya. Gutlyanskii)
Abstract. We investigate tangent spaces and metric space valued
derivatives at points of a general metric spaces. The conditions under
which two different subspaces of the metric space have isometric tangent
spaces in the common point of these subspaces are determinated.
2000 MSC. 54E35.
Key words and phrases. Metric spaces, tangent spaces.
1. Introduction. Tangent metric spaces
The recent achievements in the metric space theory are closely related
to some generalizations of the differentiation. The concept of the upper
gradient [9,10,12], Cheeger’s notion of differentiability for Rademacher’s
theorem in certain metric measure spaces [5], the metric derivative in the
studies of metric space valued functions of bounded variation [1, 4] and
the Lipshitz type approach in [8] are interesting and important examples
of such generalizations. A very interesting technical tool to develop a
theory of a differentiation in metric separable spaces is the fact that
every separable metric space admits an isometric embedding into the
dual space of a separable Banach space. It provides a linear structure,
and so a differentiation, for separable metric spaces, see for example a
rather complete theory of rectifiable sets and currents on metric spaces
in [2, 3].
These generalizations of the differentiability usually lead to nontrivial
results only for the assumption that metric spaces have “sufficiently many”
rectifiable curves. In almost all mentioned approaches we see that theories
of differentiations in metric spaces involve an induced linear structure that
is able to use the classical differentiations in the linear normed spaces.
Received 11.09.2008
ISSN 1810 – 3200. c© Iнститут математики НАН України
O. A. Dovgoshey 471
A new, intrinic, notion of differentiabililty for the mappings between
the general metric spaces was produced by O. Dovgoshey and O. Martio
in [7]. A basic technical tool in [7] is tangent spaces to an arbitrary
metric space X at a point a ∈ X that was defined as a factor space
of a family of sequences of points xn ∈ X which converge to a. This
approach makes possible to define a metric space valued derivative of
functions f : X → Y, X and Y are metric spaces, as a mapping between
tangent spaces to X at the point a and, respectively, to Y at the point
f(a). The analysis of general properties of tangent spaces and of metric
space valued derivatives is the main purpose of the present paper.
Let (X, d) be a metric space and let a be point of X. Fix some
sequence r̃ of positive real numbers rn which tend to zero. In what
follows this sequence r̃ be called a normalizing sequence. Let us denote
by X̃ the set of all sequences of points from X.
Definition 1.1. Two sequences x̃, ỹ ∈ X̃, x̃ = {xn}n∈N and ỹ =
{yn}n∈N are mutually stable (with respect to a normalizing sequence r̃ =
{rn}n∈N) if there is a finite limit
lim
n→∞
d(xn, yn)
rn
:= d̃r̃(x̃, ỹ) = d̃(x̃, ỹ). (1.1)
We shall say that a family F̃ ⊆ X̃ is self-stable (w.r.t. r̃) if every two
x̃, ỹ ∈ F̃ are mutually stable. A family F̃ ⊆ X̃ is maximal self-stable if F̃
is self-stable and for an arbitrary z̃ ∈ X̃ either z̃ ∈ F̃ or there is x̃ ∈ F̃
such that x̃ and z̃ are not mutually stable.
A standard application of Zorn’s lemma leads to the following
Proposition 1.1. Let (X, d) be a metric space and let a ∈ X. Then
for every normalizing sequence r̃ = {rn}n∈N there exists a maximal self-
stable family X̃a = X̃a,r̃ such that ã := {a, a, . . . } ∈ X̃a.
Note that the condition ã ∈ X̃a implies the equality
lim
n→∞
d(xn, a) = 0
for every x̃ = {xn}n∈N which belongs to X̃a.
Consider a function d̃ : X̃a × X̃a → R where d̃(x̃, ỹ) = d̃r̃(x̃, ỹ) is
defined by (1.1). Obviously, d̃ is symmetric and nonnegative. Moreover,
the triangle inequality for d implies
d̃(x̃, ỹ) ≤ d̃(x̃, z̃) + d̃(z̃, ỹ)
for all x̃, ỹ, z̃ ∈ X̃a. Hence (X̃a, d̃) is a pseudometric space.
472 Tangent spaces...
Definition 1.2. The pretangent space to the space X at the point a w.r.t.
a normalizing sequence r̃ is the metric identification of the pseudometric
space (X̃a,r̃, d̃).
Since the notion of pretangent space is basic for the present paper,
we remaind this metric identification construction.
Define a relation ∼ on X̃a by x̃ ∼ ỹ if and only if d̃(x̃, ỹ) = 0. Then
∼ is an equivalence relation. Let us denote by Ωa = Ωa,r̃ = ΩX
a,r̃ the set
of equivalence classes in X̃a under the equivalence relation ∼. It follows
from general properties of pseudometric spaces, see, for example, [11,
Chapter 4, Th. 15], that if ρ is defined on Ωa by
ρ(α, β) := d̃(x̃, ỹ) (1.2)
where x̃ ∈ α and ỹ ∈ β, then ρ is the well-defined metric on Ωa. The
metric identification of (X̃a, d̃) is, by definition, the metric space (Ωa, ρ).
Remark that Ωa,r̃ 6= ∅ because the constant sequence ã belongs to
X̃a,r̃, see Proposition 1.1.
Let {nk}k∈N be an infinite, strictly increasing sequence of natural
numbers. Let us denote by r̃′ the subsequence {rnk
}k∈N of the normaliz-
ing sequence r̃ = {rn}n∈N and let x̃′ := {xnk
}k∈N for every x̃ = {xn}n∈N ∈
X̃. It is clear that if x̃ and ỹ are mutually stable w.r.t. r̃, then x̃′ and ỹ′
are mutually stable w.r.t. r̃′ and
d̃r̃(x̃, ỹ) = d̃r̃′(x̃
′, ỹ′). (1.3)
If X̃a,r̃ is a maximal self-stable (w.r.t. r̃) family, then, by Zorn’s lemma,
there exists a maximal self-stable (w.r.t. r̃′) family X̃a,r̃′ such that
{x̃′ : x̃ ∈ X̃a,r̃} ⊆ X̃a,r̃′ .
Denote by inr̃′ the mapping from X̃a,r̃ to X̃a,r̃′ with inr̃′(x̃) = x̃′ for all
x̃ ∈ X̃a,r̃. It follows from (1.2) that after metric identifications inr̃′ pass
to an isometric embedding em′: Ωa,r̃ → Ωa,r̃′ under which the diagram
X̃a,r̃
in
r̃′−−−−−→ X̃a,r̃′
p
y
y
p′
Ωa,r̃
em′
−−−−−−→ Ωa,r̃′
(1.4)
is commutative. Here p and p′ are metric identification mappings, p(x̃) :=
{ỹ ∈ X̃a,r̃ : d̃r̃(x̃, ỹ) = 0} and p′(x̃) := {ỹ ∈ X̃a,r̃′ : d̃r̃′(x̃, ỹ) = 0}.
Let X and Y be two metric spaces. Recall that a map f : X → Y is
called an isometry if f is distance-preserving and onto.
O. A. Dovgoshey 473
Definition 1.3. A pretangent Ωa,r̃ is tangent if em′: Ωa,r̃ → Ωa,r̃′ is an
isometry for every r̃′.
To verify the correctness of this definition, we must prove that if
X̃
(1)
a,r̃′ and X̃a,r̃′ are two distinct maximal self-stable families such that
the double inclusion
X̃a,r̃′ ⊇
{
x̃′ : x̃ ∈ X̃a,r̃
}
⊆ X̃
(1)
a,r̃′ (1.5)
holds and em′ : Ωa,r̃ → Ωa,r̃′ is an isometry, then em′
1 : Ωa,r̃ → Ω
(1)
a,r̃′ is
also an isometry, where Ω
(1)
a,r̃′ is the metric identification of X̃
(1)
a,r̃′ . Indeed,
it is clear that if x̃ = {xn}n∈N
∈ X̃a,r̃, ỹ = {yk}k∈N
∈ X̃a,r̃′ and
lim
k→∞
d(yk, xnk
)
rnk
= 0,
then there is z̃ ∈ X̃a,r̃ with z̃′ = ỹ. Consequently, since em′ is an isometry
and (1.4) is commutative, the mapping inr′ : X̃a,r̃ → X̃a,r̃′ is surjective,
i.e.,
X̃a,r̃′ =
{
x̃′ : x̃ ∈ X̃a,r̃
}
.
Hence, by (1.5), we obtain the inclusion X̃
(1)
a,r̃′ ⊇ X̃a,r̃′ . It implies the
equality X̃
(1)
a,r̃′ = X̃a,r̃′ because X̃a,r̃′ is maximal self-stable. Hence em′
1 =
em′, so that em′
1 is an isometry.
These arguments give the following proposition.
Proposition 1.2. Let X be a metric space with a marked point a, r̃ a
normalizing sequence and X̃a,r̃ a maximal self-stable family with pretan-
gent space Ωa,r̃. The following statements are equivalent.
(i) Ωa,r̃ is tangent.
(ii) For every subsequence r̃′ of the sequence r̃ the family {x̃′ : x̃ ∈ X̃a,r̃}
is maximal self-stable w.r.t. r̃′.
(iii) A function em′ : Ωa,r̃ → Ωa,r̃′ is surjective for every r̃′.
(iv) A function in′
r : X̃a,r̃ → X̃a,r̃′ is surjective for every r̃′.
Now we introduce an equivalence relation for the classification of nor-
malizing sequences.
Definition 1.4. Let X be a metric space with a marked point a. Two
normalizing sequences r̃ and t̃ are equivalent at the point a, r̃ ≈ t̃, if the
logical equivalence
(F̃ is self-stable w.r.t. r̃) ⇐⇒ (F̃ is self-stable w.r.t. t̃)
is true for every F̃ ⊆ X̃ with ã ∈ F̃ .
474 Tangent spaces...
A normalizing sequence r̃ will be called confluented in a point a if
there exists an one-point pretangent space Ωa,r̃ (it certainly implies that
all pretangent Ωa,r̃ are one-point).
Theorem 1.1. Let (X, d) be a metric space with a marked point a and
let r̃ = {rn}n∈N
, t̃ = {tn}n∈N
be two normalizing sequences which are
equivalent at the point a. Then at least one of the following statements
holds.
(i) There is a real number c > 0 such that
lim
n→∞
rn
tn
= c. (1.6)
(ii) The sequences r̃ and t̃ are confluented in the point a.
Proof. Suppose that both sequences r̃ and t̃ are not confluented in a.
Then there are x̃ = {xn}n∈N
and ỹ = {yn}n∈N
from X̃ such that
d̃r̃(x̃, ã) = lim
n→∞
d(xn, a)
rn
> 0 and d̃t̃(ỹ, ã) = lim
n→∞
d(yn, a)
tn
> 0 (1.7)
where ã = (a, a, . . . ). If d̃r̃(ỹ, ã) > 0 or d̃t̃(x̃, ã) > 0, then we obtain
0 <
d̃r̃(ỹ, ã)
d̃t̃(ỹ, ã)
= lim
n→∞
tn
rn
<∞
or, respectively,
0 <
d̃t̃(x̃, ã)
d̃r̃(x̃, ã)
= lim
n→∞
rn
tn
<∞,
i.e., Statement (i) holds. Now observe that the equalities
d̃r̃(ỹ, ã) = d̃t̃(x̃, ã) = 0 (1.8)
lead to a contradiction because (1.7) and (1.8) imply
0 = lim
n→∞
rn
tn
= ∞.
Thus if Statement (i) does not hold, then at least one of the sequences
r̃ and t̃ is confluented. We claim that if r̃ or t̃ is confluented, then both
r̃ and t̃ are confluented. Indeed, if r̃ is confluented and we have a finite
limit
d̃t̃(ỹ, ã) = lim
n→∞
d(yn, a)
tn
6= 0 (1.9)
O. A. Dovgoshey 475
for ỹ = {yn}n∈N ∈ X̃, then d̃r̃(ỹ, ã) = 0 because t̃ ≈ r̃ and r̃ is confluented
in a. Write
y∗n :=
{
yn if n is odd
a if n is even
(1.10)
for every n ∈ N and put ỹ∗ := {y∗n}n∈N. Then we obtain d̃r̃(ỹ
∗, ã) = 0.
Thus the family
F̃ := {ỹ, ỹ∗, ã}
is self-stable w.r.t. r̃. Since r̃ ≈ t̃, this family also is self-stable w.r.t t̃.
Consequently there is a finite limit
d̃t̃(ã, ỹ
∗) = lim
n→∞
d(y∗n, a)
tn
.
Hence, by (1.9) and (1.10), we obtain
0 6= lim
n→∞
d(y∗2n+1, a)
t2n+1
= lim
n→∞
d(y∗2n, a)
t2n
= 0.
This contradiction shows that t̃ is confluented if r̃ is confluented.
Hence Statement (i) holds if Statement (ii) does not hold, and the
theorem follows.
Remark 1.1. It is clear that if there is c > 0 such that (1.6) holds, then
normalizing sequences r̃ and t̃ are equivalent at all points of an arbitrary
metric space.
Proposition 1.3. Let (X, d) be a metric space with a marked point a.
The following propositions are equivalent.
(i) The point a is an isolated point of the metric space X.
(ii) Every two normalizing sequences are equivalent at the point a.
(iii) All normalizing sequences are confluented in a.
Proof. Implications (i) ⇒ (ii) and (i) ⇒ (iii) are trivial. To prove
(ii) ⇒ (i) suppose that the relation
r̃ ≈ t̃ (1.11)
holds for every two normalizing r̃ and t̃ but there is x̃ = {xn}n∈N
∈ X̃
such that limn→∞ d(xn, a) = 0 and d(xn, a) > 0 for all n ∈ N. Let
x̃′ = {xnk
}k∈N
be an infinite subsequence of x̃ with
lim
k→∞
d(xnk
, a)
d(xk, a)
= 0. (1.12)
476 Tangent spaces...
Write
r̃ := {d(xk, a)}k∈N
, t̃ := {d(xnk
, a)}k∈N
.
Now, by the construction, both r̃ and t̃ are not confluented and, more-
over, (1.12) imply that the negation of (1.6) is true for all c > 0. Hence,
by Theorem 1.1, r̃ and t̃ are not equivalent at the point a, contrary to
(1.11). Thus (ii) ⇒ (i) is true.
If a is not an isolated point of X, then there is a sequence b̃ =
{bn}n∈N ∈ X̃ such that limn→∞ d(a, bn) = 0 and d(a, bn) 6= 0 for all
n ∈ N. Consider the normalizing sequence r̃ = {rn}n∈N with rn :=
d(a, bn). It follows immediately from (1.1) that d̃r̃(ã, b̃) = 1 where ã is
the constant sequence {a, a, . . . }. The application of Zorn’s lemma shows
that there is a maximal self-stable family X̃a,r̃ such that ã, b̃ ∈ X̃a,r̃. Then
the metric identification of the pseudometric space (X̃a,r̃, d̃) has at least
two points. Consequently we also have (iii) ⇒ (i).
2. Metric space valued derivatives.
Definition and general properties
Let (Xi, di), i = 1, 2, be metric spaces with marked points ai ∈ Xi
and r̃i = {r
(i)
n }n∈N normalizing sequences and X̃i
ai,r̃i
maximal self-stable
families with correspondent pretangent spaces Ωai,r̃i . For functions f :
X1 → X2 define the mappings f̃ : X̃1 → X̃2 as
f̃(x̃) = {f(xi)}i∈N
for x̃ = {xi}i∈N
∈ X̃1. (2.1)
Definition 2.1. A function f : X1 → X2 is differentiable w.r.t. the pair
(X̃1
a1,r̃1
, X̃2
a2,r̃2
) if the following conditions are satisfied:
(i) f̃(x̃) ∈ X̃2
a2,r̃2
for every x̃ ∈ X̃1
a1,r̃1
;
(ii) The implication
(
d̃r̃1(x̃, ỹ) = 0
)
=⇒
(
d̃r̃2(f̃(x̃), f̃(ỹ)) = 0
)
is true
for all x̃, ỹ ∈ X̃1
a1,r̃1
, where
d̃r̃1(x̃, ỹ) = lim
n→∞
d1(xn, yn)
r
(1)
n
,
d̃r̃2(f̃(x̃), f̃(ỹ)) = lim
n→∞
d2(f(xn), f(yn))
r
(2)
n
.
Remark 2.1. Note that Condition (i) of Definition 2.1 implies the
equality f(a1) = a2.
Let pi : X̃i
ai,r̃i
→ Ωai,r̃i , i = 1, 2, be metric identification mappings.
O. A. Dovgoshey 477
Definition 2.2. A function D∗f : Ωa1,r̃1 → Ωa2,r̃2 is a metric space
valued derivative of f : X1 → X2 at the point a1 ∈ X1 (or, in short,
a derivative of f) if f is differentiable w.r.t. (X̃1
a1,r̃1
, X̃2
a2,r̃2
) and the
following diagram
X̃1
a1,r̃1
X̃2
a2,r̃2
Ωa1,r̃1 Ωa2,r̃2
-f̃
?
p1
?
p2
-D∗f
(2.2)
is commutative.
In this section we establish some common properties of the metric
space valued derivatives.
Let us show that the metric space valued derivative is unique if exists.
Indeed, suppose that diagram (2.2) is commutative with D∗f = D∗
1f and
with D∗f = D∗
2f . Let β ∈ Ωa1,r̃1 . Since p1 is a surjection, there is
x̃1 ∈ X̃1
a1,r̃1
such that β = p1(x̃1). From the commutativity of (2.2) we
obtain
D∗
1(β) = D∗
1(p1(x̃1)) = p2(f̃(x̃1)) = D∗
2(p1(x̃1)) = D∗
2(β),
i.e., D∗
1 = D∗
2.
The following proposition shows that the Chain Rule remains valid
for the metric space valued derivatives.
Proposition 2.1. Let Xi be metric spaces with marked points ai ∈ Xi
and r̃i normalizing sequences and X̃i
ai,r̃i
maximal self-stable families with
pretangent spaces Ωai,r̃i , i = 1, 2, 3.
Let f : X1 → X2 and g : X2 → X3 be differentiable functions, f
w.r.t. the pair (X̃1
a1,r̃1
, X̃2
a2,r̃2
) and g w.r.t. (X̃2
a2,r̃2
, X̃3
a3,r̃3
).
Then the superposition ψ = g◦f is differentiable w.r.t. (X̃1
a1,r̃1
, X̃3
a3,r̃3
)
and
D∗(ψ) = (D∗g) ◦ (D∗f). (2.3)
Proof. The differentiability of ψ is an immediate consequence of the dif-
ferentiability of f and g, see Definition 2.1. To prove (2.3) note that
p2 ◦ f̃ = (D∗f) ◦ p1 and p3 ◦ g̃ = (D∗g) ◦ p2,
478 Tangent spaces...
see the following diagram
X̃1
a1,r̃1
X̃3
a3,r̃3
X̃2
a2,r̃2
Ωa2,r̃2
Ωa1,r̃1 Ωa3,r̃3
-ψ̃
@
@
@R
f̃
?
p1
?
p3
�
�
��g̃
?
p2
@
@
@@R
D∗g
�
�
���D∗f
-D∗ψ
(2.4)
where ψ̃ = g̃ ◦ f̃ . Consequently we have
p3 ◦ ψ̃ = p3 ◦ (g̃ ◦ f̃) = (p3 ◦ g̃) ◦ f̃ = ((D∗g) ◦ p2) ◦ f̃
= (D∗g) ◦ (p2 ◦ f̃) = (D∗g) ◦ (D∗f) ◦ p1,
that is
p3 ◦ ψ̃ = (D∗g ◦D∗f) ◦ p1.
Hence the diagram
X̃1
a1,r̃1
X̃3
a3,r̃3
Ωa1,r̃1 Ωa3,r̃3
-ψ̃
?
p1
?
p3
-(D∗g)◦(D∗f)
is commutative. The uniqueness of the derivative D∗ψ and Definition 2.2
imply (2.3).
Proposition 2.2. Let (X, d) and (Y, ρ) be metric spaces, a ∈ X and
b ∈ Y marked points in these spaces, and f : X → Y a function such
that f(a) = b. Suppose for every maximal self-stable family X̃a,r̃ ⊆ X̃
there is a maximal self-stable family Ỹb,t̃ ⊆ Ỹ such that f is differentiable
w.r.t. the pair (X̃a,r̃, Ỹb,t̃). Then f is continuous at the point a.
Proof. We may suppose that a is not an isolated point of X. Let x̃ =
{xn}n∈N ∈ X̃ be a sequence such that
lim
n→∞
d(xn, a) = 0 and d(xn, a) 6= 0
O. A. Dovgoshey 479
for all n ∈ N. Then there is a maximal self-stable family X̃a,r̃ ⊇ {ã, x̃}
where r̃ = {rn}n∈N is a normalizing sequence with rn := d(xn, a) for
n ∈ N. Hence, by the supposition, there exists a normalizing sequence
t̃ = {tn}n∈N for which the limit
lim
n→∞
ρ(f(xn), b)
tn
is finite. Consequently we have limn→∞ ρ(f(xn), b) = 0 because
lim
n→∞
tn = 0.
Hence the function f is continuous at the point a.
3. Tangent spaces to subspaces of metric spaces
Let (X, d) be a metric space with a marked point a, let Y and Z be
subspaces of X such that a ∈ Y ∩Z and let r̃ = {rn}n∈N be a normalizing
sequence.
Definition 3.1. The subspaces Y and Z are tangent equivalent at the
point a w.r.t. r̃ if for every ỹ1 = {y
(1)
n }n∈N ∈ Ỹ and every z̃1 =
{z
(1)
n }n∈N ∈ Z̃ with finite limits
d̃r̃(ã, ỹ1) = lim
n→∞
d(y
(1)
n , a)
rn
and d̃r̃(ã, z̃1) = lim
n→∞
d(z
(1)
n , a)
rn
there exist ỹ2 = {y
(2)
n }n∈N ∈ Ỹ and z̃2 = {z
(2)
n }n∈N ∈ Z̃ such that
lim
n→∞
d(y
(1)
n , z
(2)
n )
rn
= lim
n→∞
d(y
(2)
n , z
(1)
n )
rn
= 0.
We shall say that Y and Z are strongly tangent equivalent at a if Y
and Z are tangent equivalent at a for all normalizing sequences r̃.
Let F̃ ⊆ X̃. For a normalizing sequence r̃ we define a family [F̃ ]Y =
[F̃ ]Y,r̃ by the rule
(ỹ ∈ [F̃ ]Y ) ⇔ ((ỹ ∈ Ỹ ) & (∃ x̃ ∈ F̃ : d̃r̃(x̃, ỹ) = 0)). (3.1)
Note that [F̃ ]Y can be empty for some nonvoid families F̃ if the set X \Y
is “big enough”.
Proposition 3.1. Let Y and Z be subspaces of a metric space X and let
r̃ be a normalizing sequence. Suppose that Y and Z are tangent equivalent
(w.r.t. r̃) at a point a ∈ Y ∩Z. Then following statements hold for every
maximal self-stable (in Z̃) family Z̃a,r̃.
480 Tangent spaces...
(i) The family [Z̃a,r̃]Y is maximal self-stable (in Ỹ ) and we have the
equalities
[[Z̃a,r̃]Y ]Z = Z̃a,r̃ = [Z̃a,r̃]Z . (3.2)
(ii) If ΩZ
a,r̃ and ΩY
a,r̃ are metric identifications of Z̃a,r̃ and, respectively,
of Ỹa,r̃ := [Z̃a,r̃]Y , then the mapping
ΩZ
a,r̃ ∋ α 7−→ [α]Y ∈ ΩY
a,r̃ (3.3)
is an isometry. Furthermore, if ΩZ
a,r̃ is tangent, then ΩY
a,r̃ also is
tangent.
Proof. (i) Let ỹ1, ỹ2 ∈ Ỹa,r̃ := [Z̃a,r̃]Y . Then, by (3.1), there exist z̃1, z̃2 ∈
Z̃a,r̃ such that
d̃r̃(ỹ1, z̃1) = d̃r̃(ỹ2, z̃2) = 0. (3.4)
Since z̃1 and z̃2 are mutually stable, ỹ1 and ỹ2 also are mutually stable.
Consequently, Ỹa,r̃ is self-stable. The similar arguments show that [Ỹa,r̃]Z
is also self-stable. Moreover, since
[[Z̃a,r̃]Y ]Z = [Ỹa,r̃]Z ⊇ Z̃a,r̃,
the maximality of Z̃a,r̃ implies the first equality in (3.2). The second one
also simply follows from the maximality of Z̃a,r̃. It still remains to prove
that Ỹa,r̃ is a maximal self-stable subset of Ỹ . Let Ỹ m
a,r̃ be a maximal
self-stable family in Ỹ such that Ỹ m
a,r̃ ⊇ Ỹa,r̃. Then [Ỹ m
a,r̃]Z is self-stable
and [Ỹ m
a,r̃]Z ⊇ Z̃a,r̃. Since Z̃a,r̃ is maximal self-stable, the last inclusion
implies the equality [Ỹ m
a,r̃]Z = Z̃a,r̃. Using this equality and (3.2) we
obtain
Ỹ m
a,r̃ = [[Ỹ m
a,r̃]Z ]Y = [Z̃a,r̃]Y := Ỹa,r̃,
i.e., Ỹa,r̃ is maximal self-stable.
(ii) Let α ∈ ΩZ
a,r̃ and let z̃ ∈ Z̃a,r̃ such that z̃ ∈ α. It follows from
(3.1) that
[α]Y =
{
ỹ ∈ Ỹ : d̃r̃(ỹ, z̃) = 0
}
. (3.5)
The last equality implies that function (3.3) is distance-preserving. In
addition, using (3.5) we see that
[[α]Y ]Z = α and [[β]Z ]Y = β
for every α ∈ ΩZ
a,z̃ and every β ∈ ΩY
z,r̃. Consequently function (3.3) is
bijective. To prove that ΩY
a,r̃ is tangent if ΩZ
a,r̃ is tangent we can use
Statement (ii) of Proposition 1.2 and Statement (i) of Proposition 3.1.
O. A. Dovgoshey 481
Corollary 3.1. Let Y and Z be subspaces of a metric space X. Suppose
that Y and Z are tangent equivalent at a point a ∈ Y ∩ Z w.r.t. a
normalizing sequence r̃ and that there exists a unique maximal self-stable
(in Z̃) family Z̃a,r̃ ∋ ã. Then Ỹa,r̃ := [Z̃a,r̃]Y is a unique maximal self-
stable family (in Ỹ ) which contains ã.
Proof. Let Y ∗
a,r̃ ∋ ã be a maximal self-stable family in Ỹ . Then, by
Statement (i) of Proposition 3.1, [Y ∗
a,r̃]Z is maximal self-stable (in Z̃).
Since ã ∈ [Y ∗
a,r̃], we have [Y ∗
a,r]Z = Z̃a,r. Hence, by (3.2),
Y ∗
a,r = [[Y ∗
a,r]Z ]Y = [Z̃a,r̃]Y = Ỹa,r̃.
Let Y be a subspace of a metric space (X, d). For a ∈ Y and t > 0
we denote by
SYt = SY (a, t) := {y ∈ Y : d(a, y) = t}
the sphere (in the subspace Y ) with the center a and the radius t. Simi-
larly for a ∈ Z ⊆ X and t > 0 define
SZt = SZ(a, t) := {z ∈ Z : d(a, z) = t}.
Write
εa(t, Z, Y ) := sup
z∈SZ
t
inf
y∈Y
d(z, y) (3.6)
and
εa(t) = εa(t, Z, Y ) ∨ εa(t, Y, Z). (3.7)
Theorem 3.1. Let Y and Z be subspaces of a metric space (X, d) and
let a ∈ Y ∩Z. Then Y and Z are strongly tangent equivalent at the point
a if and only if
lim
t→0
εa(t)
t
= 0. (3.8)
Proof. Suppose that limit relation (3.8) holds. Let r̃ = {rn}n∈N be a
normalizing sequence and z̃ = {zn}∈N ∈ Z̃ be a sequence with a finite
limit
d̃r̃(ã, z̃) = lim
n→∞
d(a, zn)
rn
.
To find ỹ = {yn} ∈ Ỹ such that
d̃r̃(ỹ, z̃) = 0 (3.9)
482 Tangent spaces...
note that we can take ỹ = ã if d̃r̃(ã, z̃) = 0. Hence, without loss of
generality, we suppose
0 < d̃r̃(ã, z̃) = lim
n→∞
d(zn, a)
rn
. (3.10)
It follows from (3.7) and (3.8) that
lim
t→0
εa(t, Z, Y )
t
= 0. (3.11)
Inequality (3.10) implies that there is n0 ∈ N such that d(zn, a) > 0 if
n ≥ n0. Write for every n ∈ N
tn :=
{
1 if n < n0
d(zn, a) if n ≥ n0.
(3.12)
The definition of εa(t, Z, Y ) implies that for every n ∈ N there is yn ∈ Y
with
d(zn, yn) ≤ εa(tn, Z, Y ) + t2n. (3.13)
Put ỹ = {yn}n∈N where yn are points in Y for which (3.13) holds. Now
using (3.10)–(3.12) we obtain
lim sup
n→∞
d(zn, yn)
rn
≤ lim
n→∞
d(zn, a)
rn
lim sup
n→∞
d(zn, yn)
tn
≤ d̃r̃(z̃, ã) lim sup
n→∞
εa(tn, Z, Y ) + t2n
tn
= 0.
Consequently, limn→∞
d(zn,yn)
rn
= 0, i.e., (3.9) holds. Similarly, we can
prove that for every ỹ ∈ Ỹ with a finite d̃r̃(ỹ, ã) there is z̃ ∈ Z̃ such that
d̃r̃(z̃, ỹ) = 0. Hence if (3.8) holds, then Y and Z are strongly tangent
equivalent at the point a.
Suppose now that (3.8) does not hold. More precisely, we shall assume
that
lim sup
n→∞
εa(t, Z, Y )
t
> 0.
Then there is a sequence t̃ of positive numbers tn with limn→∞ tn = 0
and there is c > 0 such that for every n ∈ N there exists zn ∈ SZ(a, tn)
for which
inf
y∈Y
d(zn, y) ≥ ctn = cd(a, zn). (3.14)
O. A. Dovgoshey 483
Let us denote by z̃ the sequence of points zn ∈ Z which satisfy (3.14).
Take the sequence t̃ = {tn}n∈N as a normalizing sequence. Then, by
(3.14), we obtain
lim sup
n→∞
d(zn, yn)
tn
≥ c > 0
for every ỹ = {yn}n∈N ∈ Ỹ . Consequently, Y and Z are not strongly
tangent equivalent at the point a.
Consider now the case where Z = X. Let (X, d) be a metric space
and let a ∈ Y ⊆ X. If X̃a,r̃ ⊆ X̃ and Ỹa,r̃ ⊆ Ỹ are maximal self-stable
families and Ỹa,r̃ ⊆ X̃a,r̃, then there is a unique isometric embedding
EmY
: ΩY
a,r̃ → Ωa,r̃ such that the following diagram
Ỹa,r̃
in
−−−−−−→ X̃a,r̃
pY
y
y
p
ΩY
a,r̃
EmY−−−−−−−→ Ωa,r̃
(3.15)
is commutative. Here Ωa,r̃ and ΩY
a,r̃ are pretangent spaces correspond-
ing to X̃a,r and, respectively, to Ỹa,r̃, pY and p are appropriate metric
identification maps and in(ỹ) = ỹ for all ỹ ∈ Ỹa,r̃.
Corollary 3.2. Let (X, d) be a metric space, let Y be a subspace of X
and let a ∈ Y . The following conditions are equivalent.
(i) For every maximal self-stable X̃a,r̃ ⊆ X̃ there is a maximal self-
stable Ỹa,r̃ ⊆ Ỹ such that Ỹa,r̃ ⊆ X̃a,r̃ and the embedding EmY
:
ΩY
a,r̃ → Ωa,r̃ is an isometry.
(ii) The equality
lim
t→0
εa(t,X, Y )
t
= 0
holds.
(iii) X and Y are strongly tangent equivalent at the point a.
Proof. The equivalence (ii) ⇔ (iii) follows from Theorem 3.1. To prove
(iii) ⇒ (i) note that
[X̃a,r̃]Y = Ỹ ∩ X̃a,r̃
for every maximal self-stable X̃a,r̃ ifX and Y are strongly tangent equiva-
lent at the point a. Consequently, (iii) implies (i) because mapping (3.3)
is an isometry.
484 Tangent spaces...
Now suppose that Condition (i) is satisfied. To prove (i) ⇒ (iii) it
is sufficient to show that for every maximal self-stable X̃a,r̃ and every
x̃0 ∈ X̃a,r̃ there is ỹ0 ∈ Ỹ such that
d̃r̃(x̃0, ỹ0) = 0. (3.16)
Let α = p(x̃0). Since EmY
is an isometry, EmY
is surjective. Thus
E−1
mY
(α) 6= ∅. The last relation and
p−1
Y (E−1
mY
(α)) 6= ∅
are equivalent because pY also is surjective. Since
EmY
◦ pY = p ◦ in
and p−1(α) = {x̃ ∈ X̃a,r̃ : d̃r̃(x̃, x̃0) = 0}, we have
∅ 6= p−1
Y (E−1
mY
(α)) = in−1(p−1(α))
= in−1({x̃ ∈ X̃a,r̃ : d̃r̃(x̃0, x̃) = 0})
= {ỹ ∈ Ỹ : d̃r̃(x̃0, ỹ) = 0},
that implies (3.16) with some ỹ0 ∈ Ỹ .
Obviously, Condition (ii) of Corollary 3.2 is satisfied if Y is a dense
subset of X. Therefore, we have the following.
Corollary 3.3. Let (X, d) be a metric space and let Y be a dense sub-
space of X. Then X and Y are strongly tangent equivalent at all points
a ∈ Y , in particular, the pretangent spaces to X and to Y are pairwise
isometric for all normalizing sequences at every point a ∈ Y .
Consider now some examples.
The following result was proved in [6]. Let X = R or X = C or
X = R
+ = [0,∞) and let
d(x, y) = |x− y|
for all x, y ∈ X.
Proposition 3.2. Each pretangent space ΩX
0,r̃ (to X at the point 0) is
tangent and isometric to (X, d) for every normalizing sequence r̃.
Using Theorem 3.1 and Proposition 3.2 we can easily obtain future
examples of tangent spaces to some subspaces of the Euclidean space En.
The first example will be examined in details.
O. A. Dovgoshey 485
Example 3.1. Let F : [0, 1] → En, n ≥ 2, be a simple closed curve in
the Euclidean space En, i.e., F is continuous and F (0) = F (1) and
F (t1) 6= F (t2)
for every two distinct points t1, t2 ∈ [0, 1] with |t2−t1| 6= 1. We can write
F in the coordinate form
F (t) = (f1(t), . . . , fn(t)), t ∈ [0, 1].
Suppose that all functions fi, 1 ≤ i ≤ n, are differentiable at a point
t0 ∈ (0, 1) and
F ′(t0) = (f ′1(t0), . . . , f
′
n(t0)) 6= (0, . . . , 0).
(In the case t0 = 0 or t0 = 1 we must use the one-sided derivatives.) We
claim that each pretangent space to the subspace Y := F ([0, 1]) ⊆ En at
the point a = F (t0) is tangent and isometric to R for every normalizing
sequence r̃. Indeed, by Proposition 3.1 and by Proposition 3.2, it is
sufficient to show that Y is strongly tangent equivalent to the straight
line
Z =
{
(z1(t), . . . , zn(t)) : (z1(t), . . . , zn(t))
= F ′(t0)(t− t0) + F (t0), t ∈ R
}
at the point a = F (t0).
The classical definition of the differentiability of real functions shows
that limit relation (3.8) holds with these Y and Z. Hence, by Theo-
rem 3.1, Y and Z are strongly tangent equivalent at the point a = F (t0).
Example 3.2. Let fi : [−1, 1] → R, i = 1, . . . , n, be functions such that
f1(0) = · · · = fn(0) = c where c ∈ R is a constant. Suppose all fi have a
common finite derivative b at the point 0, f ′1(0) = · · · = f ′n(0) = b. Write
a = (0, c) and X =
n
⋃
i=1
{(t, fi(t)) : t ∈ [−1, 1]},
i.e., X is an union of the graphs of the functions fi. Let us consider X
as a subspace of the Euclidean plane E2. Then each pretangent space
X̃a,r̃ to the space X at the point a is tangent and isometric to R.
Example 3.3. Let f1, f2 be two functions from the precedent example.
Put
X =
{
(x, y) : f1(x) ∧ f2(x) ≤ y ≤ f1(x) ∨ f2(x), x ∈ [−1, 1]
}
,
i.e., X is the set of points which lie between the graphs of the functions
f1 and f2. Then each pretangent space X̃a,r̃ to X at the point a = (0, c)
is tangent and isometric to R.
486 Tangent spaces...
Example 3.4. Let α be a positive real number. Write
X =
{
(x, y, z) ∈ E3 :
√
y2 + z2 ≤ x1+α, x ∈ R
+
}
,
i.e., X can be obtained by the rotation of the plane figure {(x, y) ∈ E2 :
0 ≤ y ≤ x1+α, x ∈ R
+} around the real axis. Then each pretangent
space X̃a,r̃ to X at the point a = (0, 0, 0) is tangent and isometric to R
+.
Example 3.5. Let U ⊆ C be an open set and let F : U → En, n ≥ 2,
be an one-to-one continuous function,
F (x, y) = (f1(x, y), . . . , fn(x, y)), (x, y) ∈ U,
and let (x0, y0) be a marked point of U . Suppose that all fi are differenti-
able at the point (x0, y0) and that the rank of the Jakobian matrix of F
equals two at this point. Write
X = F (U), a = F (x0, y0).
Consider the parametrized surface X as a subspace of En. Then every
pretangent space ΩX
a,r̃ is tangent and isometric to C.
Acknowledgments. The author thanks the Department of Mathe-
matics and Statistics of the University of Helsinki for the comfortable
setting in the May–June 2008 when he began to work with the initial
version of this paper. This work also was partially supported by the
Ukrainian State Foundation for Basic Researches, Grant Ф 25.1/055.
References
[1] L. Ambrosio, Metric space valued functions of bounded variation // Ann. Scuola
Norm. Sup. Pisa Cl. Sci. (4), 17 (1990), N 3, 439–478.
[2] L. Ambrosio, B. Kircheim, Rectifiable sets in metric and Banach spaces // Math.
Ann., 318 (2000), N 3, 527–555.
[3] L. Ambrosio, B. Kircheim, Currents in mertic spaces // Acta Math. 185 (2000),
N 2, 1–80.
[4] L. Ambrosio, P. Tilli, Topics on Analysis in Metric Spaces, Cambridge University
Press, Cambridge, 2004.
[5] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces //
GAFA, Geom. func. anal. 9 (1999), 428–517.
[6] O. Dovgoshey, F. Abdullayev, M. Küçükaslan, Tangent metric spaces to convex
sets on the plane // Reports in Math., Helsinki Univ., 481 (2008), 31 p.
[7] O. Dovgoshey, O. Martio, Tangent spaces to metric spaces // Reports in Math.,
Helsinki Univ., 480 (2008), 20 p.
[8] P. Hajlasz, Sobolev spaces on an arbitrary metric space // Potential Anal. 5
(1996), 403–415.
O. A. Dovgoshey 487
[9] J. Heinonen, Lectures on Analysis on Metric Spaces, Springer-Verlag, NewYork,
2001.
[10] J. Heinonen, P. Koskela, From local to global in quasiconformal structures // Proc.
Natl. Acad. Sci. USA 93 (1996), 554–556.
[11] J. L. Kelley, General Topology, D. Van Nostrand Company, Princeton, 1965.
[12] N. Shanmugalingam, Newtonian spaces: an extention of Sobolev spaces to metric
measure spaces // Rev. Mat. Iberoamericana 16 (2000), 243–279.
Contact information
Oleksiy A.
Dovgoshey
Institute of Applied Mathematics
and Mechanics of NASU,
R. Luxemburg str. 74,
Donetsk 83114,
Ukraine
E-Mail: aleksdov@mail.ru,
dovgoshey@iamm.ac.donetsk.ua
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