UPPER ESTIMATES FOR GÂTEAUX
DIFFERENTIABILITY OF BUMP FUNCTIONS IN
ORLICZ-LORENTZ SPACES
Gâteaux Differentiability
B. Zlatanov
vol. 8, iss. 4, art. 113, 2007
B. ZLATANOV
Department of Mathematics and Informatics
Plovdiv University,
24 “Tzar Assen” str.
Plovdiv, 4000, Bulgaria
EMail: bzlatanov@gmail.com
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17 November, 2007
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Communicated by:
C.P. Niculescu
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2000 AMS Sub. Class.:
46B25, 40E30.
Key words:
Orlicz function, Orlicz–Lorentz space, Smooth bump functions.
Abstract:
Upper estimates for the order of Gâteaux smoothness of bump functions in
Orlicz–Lorentz spaces d(w, M, Γ), Γ uncountable, are obtained. The best possible order of Gâteaux differentiability in the class of all equivalent norms in
d(w, M, Γ) is found.
Received:
29 January, 2007
Accepted:
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Acknowledgements:
Research is partially supported by National Fund for Scientific Research of the
Bulgarian Ministry of Education and Science, Contract MM-1401/04.
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Contents
1
Introduction
3
2
Preliminaries
5
3
Main Result
8
Gâteaux Differentiability
4
Auxiliary Lemmas
9
B. Zlatanov
vol. 8, iss. 4, art. 113, 2007
5
Gâteaux Differentiability of Bumps in d(w, M, Γ) and d(w, p, Γ)
16
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1.
Introduction
The existence of higher order Fréchet smooth norms and bump functions and its
impact on the geometrical properties of a Banach space have been subject to many
investigations beginning with the classical result for Lp –spaces in [1] and [6]. An
extensive study and bibliography may be found in [2]. As any negative result on
the existence of Gâteaux smooth bump functions immediately applies to the problem of existence of Fréchet smooth bump functions and norms, the question arises
of estimating the best possible order of Gâteaux smoothness of bump functions in a
given Banach space. A variational technique (the Ekeland variational principle) was
applied in [2] to show that in `1 (Γ), Γ uncountable, there is no continuous Gâteaux
differentiable bump function. Following the same idea and using Stegall’s variational principle, an extension of this result to Banach spaces with uncountable unconditional basis was given in [4] and to Banach spaces with uncountable symmetric
basis in [9]. As an application in [4] it was shown that in `p (Γ), Γ uncountable, there
is no continuous p–times Gâteaux differentiable bump function when p is odd and
there is no continuous ([p] + 1)–times Gâteaux differentiable bump function in the
case p 6∈ N. This is essentially different from the case `p (N), p-odd, where equivalent
p–times Gâteaux differentiable and even uniformly Gâteaux differentiable norms are
constructed (see [10] and [8] respectively). As examples of the main result in [9],
Orlicz `M (Γ) and Lorentz d(w, p, Γ), Γ uncountable are considered and estimates
for the order of Gâteaux smoothness of bump functions are obtained. Recently a
deep result on embedding of `p spaces in Orlicz–Lorentz sequence spaces d0 (p, M )
have been found in [5]. It is shown there that `p ,→ d0 (w, M ) iff `p ,→ hM iff
p ∈ [αM , βM ]. From this result naturally arises the question of finding upper estimates for the order of Gâteaux smoothness of bump functions in Orlicz–Lorentz
spaces.
It is worthwhile to mention that results about differentiability of bump functions
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vol. 8, iss. 4, art. 113, 2007
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in `p (Γ) cannot be used directly for `M (Γ) and d(w, p, Γ). Indeed, in [3] it is proved
that `p (A) is isomorphic to a subspace of d(w, p, Γ) iff A is countable. On the other
hand `M (Γ) for M ≡ tp (1 + | log t|)q at zero, p ≥ 1, q 6= 0, contains an isomorphic
copy of `p (A) iff A is countable. The problem of embedding `p (A) or `M (A) into
d(w, M, Γ), Γ uncountable is open.
In this note we give one new application of the main result of [9] in Orlicz–
Lorentz spaces d(w, M, Γ), Γ uncountable for finding upper estimates for the order
of Gâteaux smoothness of bump functions.
Let U be an open set in a Banach space X and let f : U → R be continuous.
Following [4] we shall say that f is G0ω,k –smooth, k ∈ N in U for some ω : (0, 1] →
R+ , limt→0 t−k ω(t) = 0 if for any x ∈ U , y ∈ X the representation holds
Gâteaux Differentiability
B. Zlatanov
vol. 8, iss. 4, art. 113, 2007
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f (x + ty) = f (x) +
k
X
ti
i=1
i!
f (i) (x)(y i ) + Rfk (x, y, t),
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(i)
where f , i = 1, 2, . . . , k are i–linear bounded symmetric forms on X and
|Rfk (x,y,t)|
ω(|t|)
t→0
lim
= 0.
If U = X we use the notation G0ω,[p] instead of G0ω,[p] (X) and Gk , k ∈ N for
the set of all continuous k–times Gâteaux differentiable functions on X, for which
limt→0 |Rfk (x, y, t)|/|t|k = 0. We say that the norm k · k in X is k–times Gâteaux
differentiable if it is from the class Gk (X\{0}).
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2.
Preliminaries
We use the standard Banach space terminology from [7]. Let us recall that an Orlicz function M is an even, continuous, non-decreasing convex function such that
M (0) = 0 and limt→∞ M (t) = ∞. We say that M is a non–degenerate Orlicz
function if M (t) > 0 for every t > 0.
A weight sequence w = {wn }∞
n=1 is a positive
Pndecreasing sequence such that
w1 = 1 and limn→∞ W (n) = ∞, where W (n) = j=1 wj , for any n ∈ N.
The Orlicz–Lorentz space d(w, M, Γ) is the space of all real functions x = x(α)
defined on the set Γ, for which
)
(∞
X
I(λx) = sup
wi M (λx(αi )) < ∞
Gâteaux Differentiability
B. Zlatanov
vol. 8, iss. 4, art. 113, 2007
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i=1
for some λ > 0, where the supremum is taken over all sequences {αi }∞
i=1 of different
∗ ∞
∗
elements on Γ. There exists a sequence {αi }i=1 , such that |x(α1 )| ≥ |x(α2∗ )| ≥
∗
∗
∗
∗
· · · ≥ |x(α
Pi∞)| ≥ · · · , lim∗i→∞ x(αi ) = 0, |x(α )| = 0 if α 6= αi for i ∈ N and
I(λx) = i=1 wi M (λx(αi )). The space d(w, M, Γ), equipped with the Luxemburg
norm:
o
n
x
≤1
kxk = inf λ > 0 : I
λ
is a Banach space.
By supp x we denote the set {α ∈ Γ : x(α) 6= 0}.
The symbol eγ , γ ∈ Γ will stand for the unit vectors.
If M (u) = up , 1 ≤ p < ∞ then d(w, M, Γ) is the Lorentz space d(w, p, Γ). If
wi = 1 for every i ∈ N then d(w, M, Γ) is the Orlicz space `M (Γ). In this case we
f(x)
use the notation I(x) = M
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To every Orlicz function M the following numbers are associated:
M (uv)
αM = sup p > 0 : sup p
<∞ ,
0<u,v≤1 u M (v)
M (uv)
βM = inf q > 0 : inf
>0 .
0<u,v≤1 uq M (v)
We consider only spaces generated by an Orlicz function M satisfying the ∆2 –
condition at zero, i.e., βM < ∞, which implies of course that
(2.1)
Gâteaux Differentiability
B. Zlatanov
vol. 8, iss. 4, art. 113, 2007
M (uv) ≥ uq M (v), u, v ∈ [0, 1]
for some q > βM (see [7]).
Finally we mention that the unit vectors {eγ }γ∈Γ form a symmetric basis of
d(w, M, Γ) with symmetric constant 1, which is boundedly complete [5], [7].
For a function g : (0, 1] → R+ denote:
M (uv)
dM (g) = sup
: u, v ∈ (0, 1] .
g(u)M (v)
Let us recall a well known definition. Let X haveP
symmetric basis {eγ }γ∈Γ with
a symmetric constant 1 and P
let z ∈ X, z 6= 0, z = ∞
i=1 ui eγi , γi 6= γj for i 6= j.
∞
∞
A sequence {zk }k=1 , zk = i=1 ui eαi,k , αi,k 6= αj,l for (i, k) 6= (j, l), αi,k ∈ Γ is
called a block basis generated by the vector z.
We will apply a general result for upper estimates for the order of Gâteaux smoothness of bump functions in a Banach space with a symmetric, boundedly complete
basis with a symmetric constant 1, obtained in [9].
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Theorem 2.1. [9] Let X be a Banach space, let {eγ }γ∈Γ , ]Γ > ℵ0 be a symmetric,
boundedly complete basis in X with a symmetric constant 1 and let:
n
X
1
lim zj n− k = 0
n→∞ j=1
for every z ∈ X.
Let ω : [0, 1] → R+ be such that for every x ∈ X there exist y ∈ X, suppy ∩
suppx = ∅ and a sequence tn & 0, which satisfy the inequality
kx + tn yk − kxk ≥ ω(tn ),
Gâteaux Differentiability
B. Zlatanov
vol. 8, iss. 4, art. 113, 2007
n ∈ N.
Then in X there is no continuous:
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(i) G0ω,k –smooth bump when ω(t) = o(tk );
(ii) G0ω,k+1 –smooth bump when ω(t) = o(tk+1 ), k–even;
(iii) k–times Gâteaux differentiable bump if ω(t) = tk ;
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(iv) (k + 1)–times Gâteaux differentiable bump if ω(t) = tk+1 , k–even.
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3.
Main Result
Theorem 3.1. Let M be an Orlicz function. If f is a continuous k–times Gâteaux
differentiable bump function in d(w, M, Γ), then
(
[αM ],
dM (t[αM ] ) < ∞
k ≤ EM =
αM − 1, αM ∈ N, dM (tαM ) = ∞.
Gâteaux Differentiability
B. Zlatanov
vol. 8, iss. 4, art. 113, 2007
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4.
Auxiliary Lemmas
To apply Theorem 2.1 for d(w, M, Γ) we need the following lemmas.
Lemma 4.1. Let p ≥ 1 and let M be an Orlicz function satisfying the conditions
limt→0 Mtp(t) = 0, dM (tp ) = c < ∞. Then every block basis {zj }∞
j=1 of the unit
vector basis {eγ }γ∈Γ in d(w, M, Γ), generated by one vector, satisfies
n
X
1
lim zj n− p = 0.
n→∞ j=1
P
Proof. Let z = ∞
Γ). Let {ej,i }∞
i=1 , j ∈ N be disjoint subsets
i=1 ui eγi ∈ d(w, M,
P∞
M (t)
of {eγ }γ∈Γ . Then we define zj =
. It follows that
i=1 ui ej,i . Let µ(t) =
tp
lim
Pnt→0 µ(t) = 0 and µ(t1 ) ≤ cµ(t2 ) for every 0 < t1 < t2 ≤ 1. Let λn (z) =
j=1 zj . Then
∞
ni
X
X
I(λn (z)) =
wj M (u∗i ).
i=1 j=n(i−1)+1
ni
X
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vol. 8, iss. 4, art. 113, 2007
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For every ε > 0 there exists m ∈ N such that
∞
X
Gâteaux Differentiability
wj M (u∗i )
i=m+1 j=n(i−1)+1
By the definition of the function µ it follows that
∞
∞
ni
X
X
X
u∗i
∗ p
wj |ui | µ
= kλn (z)kp
kλn (z)k
i=1
i=1
j=n(i−1)+1
p
= kλn (z)k .
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ε
< .
2c
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ni
X
j=n(i−1)+1
wj M
u∗i
kλn (z)k
Using the inequality
X
∞
λn (z)
1=I
=
kλn (z)k
i=1
ni
X
j=n(i−1)+1
wj M
u∗i
kλn (z)k
≥
n
X
wj M
j=1
u∗1
kλn (z)k
we get that limn→∞ kλn (z)k−1 = 0.
For every m ∈ N we have
m
ni
m
X
X
wj |u∗i |p
u∗i
u∗i
w1 + w2 . . . wn X ∗ p
µ
≤
|ui | µ
.
n
kλn (z)k
n
kλn (z)k
i=1
i=1
Gâteaux Differentiability
B. Zlatanov
vol. 8, iss. 4, art. 113, 2007
j=n(i−1)+1
Because limj→∞ wj = 0 it follows that for every ε > 0 and every m ∈ N there exists
N ∈ N such that for any n ≥ N holds
ni
m
X
X
wj |u∗i |p
u∗i
ε
µ
≤ .
n
kλn (z)k
2
i=1
j=n(i−1)+1
On the other hand for all n ∈ N such that kλn (z)k−1 ≤ 1 we can write the chain of
inequalities
ni
∞
ni
∞
X
X
X
X
wj |u∗i |p
u∗i
wj |u∗i |p
µ
≤c
µ(u∗i )
n
kλ
(z)k
n
n
i=m+1
i=m+1
j=n(i−1)+1
j=n(i−1)+1
≤c
∞
X
ni
X
i=m+1 j=n(i−1)+1
ε
≤
.
2n
wj
M (u∗i )
n
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Therefore for every ε > 0 and n ≥ N we have
∞
kλn (z)kp X
=
n
i=1
and thus
ni
X
j=n(i−1)+1
wj |u∗i |p
µ
n
u∗i
kλn (z)k
≤
ε
ε
+
<ε
2 2n
n
X
1
lim zj n− p = 0
n→∞ Gâteaux Differentiability
B. Zlatanov
j=1
vol. 8, iss. 4, art. 113, 2007
Lemma 4.2. Let dM (ω) = ∞ then for any x ∈ d(w, M, Γ) there exist y ∈ d(w, M, Γ)
with supp y ∩ supp x = ∅ and a sequence tn & 0 such that
(4.1)
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kx + tn yk ≥ kxk + cω(tn )
for some constant c > 0 and any n ∈ N.
Proof. We note first that
ω(t)
= 0.
t→0
t
If x = 0, choose sequence tn & 0 such that limn→∞ ω(tn )/tn = 0. Then (4.1) holds
trivially for any y 6= 0 with c = kyk > 0.
WLOG suppose that P
M (1) = 1.
Fix an arbitrary x = ∞
n=1 xn eγn ∈ d(w, M, Γ) and kxk = 1. Just for simplicity
of notation we will assume that |x1 | ≥ |x2 | ≥ · · · ≥ |xn | ≥ · · · .
We will choose sequences tn & 0 and vn & 0 inductively:
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lim inf
1. t1 = v1 = u1 = 1, k0 = 0, k1 = 1.
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2. Find k2 > k1 , k2 ∈ N such that
1
M (t1 v1 )
< M (v1 ) and M (xi ) <
P
k
2
2
21 j=k1 +1 wj
for i ≥ k2 .
Find t2 < t1 , v2 < v1 such that
Gâteaux Differentiability
M (t2 v2 )
> 22
ω(t2 )M (v2 )
and M (v2 ) <
1
22
Pk2
j=k1 +1
wj
.
3. Find k3 > k2 , k3 ∈ N such that
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1
M (t2 v2 )
< M (v2 ) and M (xi ) <
P
k
3
2
22 j=k2 +1 wj
for i ≥ k3 .
Find t3 < t2 , v3 < v2 such that
M (t3 v3 )
> 23
ω(t3 )M (v3 )
and
M (v3 ) <
23
Pk3
j=k2 +1 wj
.
for i ≥ kn .
j=kn−1 +1
wj
< M (vn−1 ) and M (xi ) <
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4. Find kn > kn−1 , kn ∈ N such that
1
Pkn
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1
If we have chosen tn−1 , vn−1 and kn−1 then
2n−1
B. Zlatanov
vol. 8, iss. 4, art. 113, 2007
M (tn−1 vn−1 )
2
Find tn < tn−1 , vn < vn−1 such that
M (tn vn )
> 2n
ω(tn )M (vn )
and M (vn ) <
1
.
P
k
n
2n j=kn−1 +1 wj
For a sequence {An }∞
n=1 of finite disjoint subsets of Γ, such that An ∩supp x = ∅,
]An = kn − kn−1 , put
Gâteaux Differentiability
yn = vn
X
eγ
and y =
∞
X
B. Zlatanov
yn .
vol. 8, iss. 4, art. 113, 2007
n=1
γ∈An
Obviously
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I(y) =
∞
X
kn
X
Contents
wj M (vn )
n=1 j=kn−1 +1
= w1 M (v1 ) +
∞
X
M (vn )
n=2
≤1+
∞
X
n=2
kn
X
wj
j=kn−1 +1
1
< ∞,
2n
≥
X
j=1
kn+2
wj M (xj ) +
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I(x + tn y) − I(x) ≥ I(x + tn yn ) − I(x)
kn+1
II
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which ensures y ∈ d(w, M, Γ). We have supp (x + tn y) = supp x ∪ (∪∞
n=1 An ) for
any t 6= 0 and therefore
(4.2)
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X
j=kn+1 +1
wj M (tn vn )
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+
∞
X
wj M (xj+kn+1 −kn+2 ) −
kn+2
X
kn+2
+
X
wj −
j=kn+1 +1
∞
X
wj M (xj )
j=1
j=kn+2 +1
= M (tn vn )
∞
X
wj M (xj )
j=kn+1 +1
wj (M (xj+kn+1 −kn+2 ) − M (xj ))
Gâteaux Differentiability
B. Zlatanov
j=kn+2 +1
vol. 8, iss. 4, art. 113, 2007
kn+2
X
1
≥ M (tn vn )
wj
2
j=k
+1
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n+1
kn+2
X
1
≥ 2n ω(tn )M (vn )
wj
2
j=k
+1
n+1
≥ 2n−1 ω(tn )
M (vn )
n+1
2 M (vn+1 )
≥
ω(tn )
.
4
Remove as many elements of the sequence {tn }∞
n=1 as necessary to obtain
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0 < dn = kx + tn yk − 1 ≤ 1
and keep the same notation for the remaining sequence. Now (2.1) implies
x + tn y
(4.3)
−1
I(x + tn y) − I(x) = I kx + tn yk
kx + tn yk
≤ kx + tn ykq − 1
= (1 + dn )q − 1 ≤ q2q−1 dn ,
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for some q > βM .
Combining (4.2) and (4.3), we obtain
kx + tn yk − 1 ≥ cω(tn ),
1
where c = q2q+1
.
Now let x 6= 0 be arbitrary. Find y such that supp y ∩ supp x = ∅ and
x
kxk − tn y − 1 ≥ cω(tn ). Obviously for y = kxky we have
kx + tn yk − kxk ≥ ckxkω(tn ).
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vol. 8, iss. 4, art. 113, 2007
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5.
Gâteaux Differentiability of Bumps in d(w, M, Γ) and d(w, p, Γ)
Theorem 5.1. Let M be an Orlicz function and ω : (0, 1] → R+ , dM (ω) = ∞.
(i) If αM 6∈ N then there is no continuous G0ω,[αM ] –smooth bump function in
d(w, M, Γ);
(ii) If αM ∈ N then there is no continuous G0ω,αM –smooth bump function, provided
dM (tαM ) < ∞ in d(w, M, Γ) and there is no continuous G0ω,αM −1 –smooth
bump function, provided dM (tαM ) = ∞ in d(w, M, Γ).
Proof. The proof in all cases is straightforward, applying Lemma 4.1 for appropriate
p, Lemma 4.2 and Theorem 2.1.
Proof of Theorem 3.1. The proof in the two cases is straightforward, applying Theorem 5.1.
It is well known that in a Banach space X a norm of some order of smoothness
generates a bump function with the same order of smoothness (see e.g. [2]), therefore
the next corollary is a direct consequence of Theorem 3.1
Corollary 5.2. Let M be an Orlicz function. If | · | is an equivalent norm in
d(w, M, Γ), which is k–times Gâteaux differentiable then k ≤ EM .
As a consequence of Theorem 5.1 and Theorem 3.1 we get for M (t) = tp , p ≥ 1
the results from [9].
P
Corollary 5.3 ([9, Theorem 3]). Let p ≥ 1, wn & 0, ∞
n=1 wn = ∞ and ω :
+
p
(0, 1] → R be such that ω(t) = o(t ). Then there is no continuous G0ω,[p] –smooth
bump function in d(w, p, Γ).
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vol. 8, iss. 4, art. 113, 2007
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Proof. Indeed in this case αM = p and dtp (ω) = ∞. If p 6∈ N then by Theorem 5.1
i), it follows that there is no continuous G0ω,[p] –smooth bump in d(w, M, Γ). If p ∈ N
then dtp (tp ) = 1 < ∞ and by Theorem 5.1 ii), there is no continuous G0ω,p –smooth
bump in d(w, M, Γ).
P
Corollary 5.4 ([9, Corollary 2]). Let p ≥ 1, wn & 0, ∞
n=1 wn = ∞. If f is a
continuous k–times Gâteaux differentiable bump function in d(w, p, Γ), then k ≤ [p].
Gâteaux Differentiability
Proof. In this case it is obvious that dtp (t[p] ) < ∞ and dtp (tp ) < ∞. Therefore by
Theorem 3.1 it follows that k ≤ [p].
B. Zlatanov
vol. 8, iss. 4, art. 113, 2007
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References
[1] N. BONIC AND J. FRAMPTON, Smooth functions on Banach manifolds, J.
Math. Mech., 15 (1966), 877–898.
[2] R. DEVILLE, G. GODEFROY AND V. ZIZLER, Smoothness and Renormings
in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol 64, Logman Scientific and Technical, Harlow/New York, 1993.
Gâteaux Differentiability
B. Zlatanov
[3] F. HERNANDEZ AND S. TROYANSKI, On representation of uncountable
symmetric basic sets and its applications, Studia Math., 107 (1993), 287–304.
vol. 8, iss. 4, art. 113, 2007
[4] F. HERNANDEZ AND S. TROYANSKI, On Gâteaux differentiable bump functions, Studia Math., 118 (1996), 135–143.
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[5] A. KAMINSKA AND Y. RAYNAUD, Isomorphic `p –subspaces in Orlicz–
Lorentz sequence spaces, Proc. Amer. Math. Soc., 134 (2006), 2317–2327.
[6] J. KURZWEIL, On approximation in real Banach space, Studia Math., 14
(1954), 213–231.
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