Series representation of compact linear operators in Banach spaces

vol. 56, no. 1 (2016), 17–27

Series representation of compact linear operators in Banach spaces

David E. Edmunds and Jan Lang

Summary. Let p ∈ (1, ∞) and I = (0, 1); suppose that T ∶ L

(I) → L

(I) is a compact linear map with trivial kernel and range dense in L

(I). It is shown that if the Gelfand numbers of T decay sufficiently quickly, then the action of T is given by a series with calculable coefficients. The special properties of L

(I) enable this to be established under weaker conditions on the Gelfand numbers than in earlier work set in the context of more general spaces.

Keywords Eigenvalues;

Banach spaces;

compact operators;

nuclear maps;

Gelfand numbers

MSC 2010

15A18; 46B50; 47B38;

47A58; 47A60 Received: 2016-04-05, Accepted: 2016-05-25

1. Introduction

In the last few years a good deal of work has been done on the representation in series form of compact linear maps T acting between Banach spaces X and Y . When these are Hilbert spaces, it is classical that such a representation is always possible, but outside the Hilbert space framework the problems presented by the lack of orthogonality present dif- ficulties that so far have been surmounted only in special circumstances: we refer to [1]

for a detailed description of such work. A particular result of this type is that given in [4], where it is shown that if X and Y are uniformly convex, uniformly smooth and infinite

David E. Edmunds, Department of Mathematics, University of Sussex, Pevensey I, Brighton, BN1 9QH, United Kingdom (e-mail: davideedmunds@aol.com)

Jan Lang, Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue, Columbus, OH 43210-1174, USA (e-mail: lang@math.osu.edu)

DOI 10.14708/cm.v56i1.1139 © 2016 Polish Mathematical Society

dimensional, while T has trivial kernel, range dense in Y and Gelfand numbers c n (T ) that decay sufficiently quickly (c n (T ) ⩽ 2 ⁻ⁿ⁺¹ /(2 ⁿ − 1) will do), then the action of T is given by

T x = ∑ _n ⟨x , α _n ^Y ⟩ _X y n , x ∈ X ,

where α ^Y _n ∈ X ^∗ and y n ∈ Y for each n. The terms in this series originate in the construction of a decreasing sequence of linear subspaces X n of X with finite codimension and trivial in- tersection; the norm of the restriction of T to X n is attained at x n and y n = T x n ; moreover, the coefficients ⟨x , α ^Y _n ⟩ X are recursively calculable by using projections and polar sets.

In this paper we show that the assumption concerning the decay of the Gelfand num- bers c n (T ) can be weakened if it is assumed that X = Y = L p (I), where p ∈ (1, ∞) and I = (0, 1) ⊂ R. The arguments hinge on a striking result of Rolewicz [ 11] that if V is any subspace of L p (I) with codimension 1, then there is a linear projection of L p (I) onto V with minimal norm; and this minimal norm is independent of V . Our claim follows from this together with the fact, established by Franchetti [5], that every polar projection is mi- nimal. The difficulties encountered even in the context of L p spaces make it tempting to speculate that power-type decay of the Gelfand numbers would not be sufficient to ensure the series representation of T considered here.

2. Preliminaries

The Banach spaces X with which we shall be concerned in this paper are assumed to be real, uniformly convex, uniformly smooth and infinite-dimensional, with norm ∥⋅∥ _X and dual X ^∗ . The closed unit ball of X is denoted by B X ; the value of x ^∗ ∈ X ^∗ at x ∈ X will be denoted by ⟨x , x ^∗ ⟩ _X . If Y is another such space, B(X , Y ) will represent the family of all bounded linear maps from X to Y , and we shall write B(X) instead of B(X , X). When M , N are closed linear subspaces of X , X ^∗ respectively, their polar sets are

M ⁰ = {x ^∗ ∈ X ^∗ ∶ ⟨x , x ^∗ ⟩ _X = 0 for all x ∈ M}

and

0 N = {x ∈ X ∶ ⟨x , x ^∗ ⟩ _X = 0 for all x ^∗ ∈ N }.

The uniform smoothness of X implies that ∥⋅∥ _X is Gâteaux-differentiable on X ∖ {0};

the Gâteaux derivative ̃ J X (x ) ∶= grad ∥x ∥ _X of ∥⋅∥ _X at x ∈ X ∖ {0} is the unique element of X ^∗ such that

∥̃ J X (x )∥ _X

= 1 and ⟨x , ̃ J X (x )⟩ _X = ∥x ∥ _X .

A gauge function is a map µ∶ [0, ∞) → [0, ∞) that is continuous, strictly increasing and such that µ(0) = 0 and lim _t→∞ µ(t) = ∞; the map J X ∶ X → X ^∗ defined by

J X (x ) = µ(∥x ∥ _X )̃ J X (x ), x ∈ X ∖ {0}, J X (0) = 0

is called a duality map on X with gauge function µ. Note that for all x ∈ X ,

⟨x , J X (x )⟩ _X = ∥J X (x )∥ _X

∥x ∥ X , ∥J X (x )∥ _X

= µ(∥x ∥ X ).

When X = L p (1 < p < ∞) the duality map J corresponding to the gauge function µ, where µ(t) = t ^p−1 , satisfies J( f ) = ∣ f ∣ ^p−1 sgn f .

Now we recall that the modulus of convexity of X , δ X ∶ [0, 2] → [0, 1] is defined by δ X (ε) = inf {1 −

∥x + y∥ X

2

∶ x , y ∈ X , ∥x ∥ X , ∥ y∥ X ⩽ 1, ∥x − y∥ X ⩾ ε}.

We say that X is uniformly convex if δ X (ε) > 0 for all ε ∈ (0, 2]. For ε ∈ [0, 2], we put ρ X (ε) = sup{

∥x + y∥ X + ∥x − y∥ X

2

− 1 ∶ ∥x ∥ X = 1, ∥ y∥ X = ε}.

The function ρ X is called the modulus of smoothness of X . We say X is a uniformly smooth Banach space if lim ε→0 (ρ X (ε)/ε) = 0. Note that for any Hilbert space H we have that

ρ H (ε) = (1 + ε ² ) ^1/2 − 1, δ H (ε) = 1 − (1 − ( ε 2 )

2

)

1/2

, ε ∈ [0, 2],

while for any Banach space X : δ X (ε) ⩽ δ H (ε); and it can be shown (see [9, page 63]) that for each ε ∈ [0, 2] we have ρ H (ε) ⩽ ρ X (ε).

For the classical L p spaces, the asymptotic behavior of the moduli is given by the following formulas (see [6, p. 244] and [8, p. 243]):

δ L

(ε) ≍ ε ² and ρ L

(t) ≍ t ^p if 1 < p ⩽ 2, (1) δ L

(ε) ≍ ε ^q and ρ L

(t) ≍ t ² if 2 ⩽ q < ∞,

as ε → 0 and t → 0, respectively.

We say that a Banach space X is strictly convex if and only if δ X (2) = 1 (i.e., the boundary of the unit ball contains no line segments).

We now summarise some of the results obtained in the last few years concerning the representation of compact linear operators acting between Banach spaces; a connected ac- count of this work is given in [1]. Let X , Y be real Banach spaces that are uniformly convex and uniformly smooth, and suppose that X and Y are equipped with duality maps J X , J Y

corresponding to gauge functions µ X , µ Y that are normalised so that µ X (1) = µ Y (1) = 1;

assume further that T ∈ B(X , Y ) is compact with trivial kernel and range dense in Y . Then there exists x ₁ ∈ X , with ∥x 1 ∥ _X = 1, such that ∥T ∥ = ∥T x 1 ∥ _Y and

T ^∗ J Y T x 1 = ν 1 J X x 1 , ν 1 = ∥T ∥ µ Y (∥T ∥) .

Write X 1 = X , M 1 = sp {J X x ₁ } , X 2 = ⁰ M ₁ , N 1 = sp{J Y T x ₁ } , Y 2 = ⁰ N ₁ and λ 1 = ∥T ∥ . Let

T ₂ be the restriction of T to X ₂ ∶ then T ₂ is a compact linear map of X ₂ to Y ₂ , and if it is

not the zero operator the procedure can be repeated to give the existence of x ₂ ∈ X 2 , with

∥x 2 ∥ _X = 1, such that

T ₂ ^∗ J Y

T ₂ x ₂ = ν 2 J X

x ₂ , ν ₂ = λ 2 µ Y (λ 2 ) , λ 2 = ∥T x 2 ∥ _Y = ∥T 2 ∥ .

In this way we obtain elements x ₁ , . . . , x n of X , each with unit norm, subspaces M ₁ , . . . , M n

of X ^∗ and N ₁ , . . . , N n of Y ^∗ , where

M k = sp{J X x ₁ , . . . , J X x k }, N k = sp{J Y T x ₁ , . . . , J Y T x k }, k = 1, . . . , n,

and decreasing families X 1 , . . . , X n and Y 1 , . . . , Y n of subspaces of X and Y , respectively, given by

X k = ⁰ M k−1 , Y k = ⁰ N k−1 , k = 2, . . . , n.

For each k ∈ {1, . . . , n}, T maps X k into Y k , x k ∈ X k and, setting T k ∶= T ↿ X

k

, λ k =

∥T k ∥ , ν k = λ k µ(λ k ), we have

⟨T x , J Y T x k ⟩ _Y = ν k ⟨x , J X x k ⟩ _X for all x ∈ X k .

The process stops with λ n , x n and X _n+1 if and only if the restriction of T to X _n+1 is the zero operator while T n ≠ 0. If T has infinite rank, then {λ n } is an infinite sequence that converges monotonically to zero, and

∞

⋂

n=1

X n = ker T = {0}.

Henceforth we shall assume that T has infinite rank.

A semi-inner product (⋅, ⋅) _X may be defined on X by

(x , h) _X = ∥x ∥ _X ⟨h, ̃ J X x ⟩ _X , x ≠ 0, (0, h) _X = 0.

With respect to this, the x k have the semi-orthogonality property (x r , x s ) _X = δ _{r ,s} if r ⩽ s,

where δ _{r ,s} is the Kronecker delta. Moreover, the elements y n ∶= T x n / ∥T x n ∥ have the same property:

( y r , y s ) _Y = δ r ,s if r ⩽ s.

It is useful to describe these results in terms of the notion of orthogonality given by James [7], using the notation of [2] and [3]. An element x ∈ X is said to be j-orthogonal (or orthogonal in the sense of James) to y ∈ X, and we write x ⊥ ^j y, if

∥x ∥ _X ⩽ ∥x + t y∥ _X for all t ∈ R.

If x is j-orthogonal to every element of a subset W of X , it is said to be j-orthogonal to W , written x ⊥ ^j W . A subset W ₁ of X is j-orthogonal to W ₂ ⊂ X , written W 1 ⊥ ^j W ₂ , if w ₁ ⊥ ^j w ₂ for all w ₁ ∈ W 1 and all w ₂ ∈ W 2 . In general, j-orthogonality is not symmetric; and in fact, x ⊥ ^j y for all x , y ∈ X if and only if X is a Hilbert space. The semi-inner product mentioned above is naturally connected to j-orthogonality: if x , h ∈ X , then x ⊥ ^j h if and only if (x , h) _X = 0. The elements x n and y n have the properties that

x r ⊥ ^j x s and y r ⊥ ^j y s if r < s, and

x r ⊥ ^j X r and y r ⊥ ^j Y r for all r . Now define

Z _n ^X = sp {x 1 , . . . , x n } , Z _n ^Y = sp { y 1 , . . . , y n } , n ∈ N, and introduce the family of maps

S _k ^X = S ^{X ,T} _k ∶ X → Z _k−1 ^X , k ⩾ 2

determined by the condition that x − S ^{X ,T} _k x ∈ X k for all x ∈ X . It can be shown that S _k ^X is uniquely given by

S _k ^X (x ) = ∑

k−1

j=1 ξ ^X _j (x )x j , where

ξ ^X _j (x ) = ⟨x −

j−1

∑

i=1

ξ ^X _i (x )x i , J X x j ⟩

X if j ⩾ 2, ξ ₁ ^X (x ) = ⟨x , J X x 1 ⟩ _X . Maps S _k ^Y are defined in an analogous fashion:

S _k ^Y ( y) =

k−1

∑

j=1

ξ ^Y _j ( y) y j , with natural definition of the ξ ^Y _j .

Since ξ ^X _j is a linear functional, S ^X _k is linear; in fact it is a projection of X onto Z ^X _k−1 . We summarise some useful features of these maps as follows.

2.1. Lemma. The spaces X and X ^∗ have the following direct sum decompositions:

X = X k ⊕ Z _k−1 ^X , X ^∗ = M _k−1 ⊕ (Z _k−1 ^X )

0

for each k ⩾ 2.

The maps S _k ^X , (S ^X _k )

∗

are respectively linear projections of X onto Z _k−1 ^X and X ^∗ onto M _k−1 . For all k ∈ N,

X k = X k+1 ⊕ sp {x k } .

Corresponding statements hold for Y .

For a proof, see [4, Lemma 2].

Next, let P _n ^X be the projection of X onto X n (n ∈ N) , by which we mean that P ^{n X} ∶ X → X n takes x ∈ X to the point in X n nearest to x ; projections P _n ^Y ∶ Y → Y n are defined analogously.

2.2. Lemma. Let n ∈ N ∖ {1}, x ∈ X and y ∈ Y . Then

x − S _n ^X (x ) = P _n ^X P _n−1 ^X . . . P ₂ ^X (x ) and y − S ^Y _n ( y) = P _n ^Y P _n−1 ^Y . . . P ₂ ^Y ( y).

This is established in [4, Lemma 4]. Note that when n = 2 we have x − S ₂ ^X (x ) = P ₂ ^X (x ),

which implies that P ₂ ^X is a linear projection onto X ₂ . In general, when n > 2 the projections P _n ^X are nonlinear, but Lemma 2.2 shows that P _n ^X P _n−1 ^X . . . P ₂ ^X is a linear projection of X onto X n and that the restriction of P _n ^X to X n−1 is a linear projection onto X n .

The Gelfand numbers c n (T ) of T are defined by

c n (T ) = inf {∥T J ^X _M ∥ ∶ codim M < n}, n ∈ N,

where J _M ^X is the natural embedding of the closed linear subspace M of X into X . In [4] it is shown that

(2 ⁿ − 1)

−1

λ n ⩽ c n (T ) ⩽ λ n , n ∈ N. (2)

This estimate is essential for our main result and is a quite simple consequence of the following Lemma which is proved in [4]:

2.3. Lemma. Let n ∈ N and let K ⁿ be the convex hull of x ₁ , . . . , x n , where the x i are defined as above. Then

(2 ⁿ − 1)

−1

⩽ inf {∥x ∥ ∶ x ∈ K n } ⩽ 1. (3)

The same holds with each x i replaced by y i .

Note that the lower bound in (3), which is true for all Banach spaces, is obtained by using quite crude estimates. Since the above estimate is essential for our main result one can consider that there is a possibility of improvement of the lower estimate by using properties of particular spaces, such as L p (1 < p < ∞).

We will try to achieve it by modifying the proof of the above lemma (from [4]) under the condition that X = L p , when we are equipped with (1).

Let x ∈ K n . Then x = ∑ ⁿ _i=1 α i x i for some non-negative α i such that ∑ ⁿ _i=1 α i = 1. Since

∑

n

i=2 α i x i ∈ X 2 and α 1 x 1 ⊥ ^j X 2 ,

α ₁ = ∥α ₁ x ₁ ∥ X ⩽ ∥α ₁ x ₁ +

n

∑

i=2

α i x i ∥

X = ∥x ∥ _X . (4)

Using (1) (when X = L p ) there exists a constant C X > 0 such that

∥x i ∥ X + C X ε ^p ⩽ ∥x i + εx i+1 ∥ X

and this yields

α i + C X α

p

j ⩽ ∥α i x i + α j x j ∥ X when i < j.

Moreover, we obtain an improvement of (4):

α j + C X α ^p _j+1 ⩽ α j + C X (α j+1 + C X (α j+3 + C X (. . . C X (α k−1 + α

p

k ) ^p . . .) ^p ) ^p )

p

⩽ ∥

k

∑

i= j

α i x i ∥

X . Thus

α 1 + C X α

p 2 ⩽ ∥

n

∑

i=1

α i x i ∥

X = ∥x ∥ X . As ∑ ⁿ _i=3 α i x i ∈ X 3 and α 2 x 2 ⊥ ^j X 3 ,

α ₂ + C X α

p

3 − α ₁ = ∥

n

∑

i=2

α i x i ∥

X − ∥α ₁ x ₁ ∥ _X

⩽ ∥

n

∑

i=1

α i x i ∥

X = ∥x ∥ _X . In the same way it follows that for each l ∈ N, with l < n,

α l + C X α _{l +1} ^p −

l −1

∑

i=1

α i ⩽ ∥

n

∑

i=1

α i x i ∥

X . Thus

inf max (α ₁ + C X α

p

2 , α ₂ + C X α

p

3 − α ₁ , α ₃ + C X α

p

4 − α ₂ − α ₁ , . . . ) ⩽ inf

x∈K

∥x ∥ _X , where the infimum on the left-hand side is taken over all α i ⩾ 0 such that ∑ ⁿ _i=1 α i = 1.

By the same arguments as in the proof of [4, Lemma 5] we can see that the infimum is attained when all the entries coincide:

α ₁ + C X α ^p

2 = α 2 + C X α ^p

3 − α 1 , . . .

and so on. Then we obtain the following non-linear system of n equations:

2α i = α i+1 − C X α

p

i+1 + C X α

p

i+2 for 1 ⩽ i ⩽ n − 2 2α _n−1 = α n − C X α

p n

1 =

n

∑

i=1

α i .

The first n − 2 equations of this system can be also viewed as a non-linear difference equ- ation and could be written as

(α _i+1 − α i ) + C X (α

p i+2 − α

p

i+1 ) − α i = 0.

In the case when C X = 0 (studied in [4]) the solution of the above system is α i = 2 ⁱ /(2 ⁿ −1) but when C X > 0 it is not obvious what is the solution of the system as one can not readily solve the above non-linear difference equation.

Numerical computation indicates that for large n, solutions of the system behave like 2 ⁱ⁻ⁿ . This suggests that detailed knowledge of the modulus of convexity of the L p spaces may not essentially improve the lower estimates in (3) and (2), even when p is very near to 2.

To conclude this preliminary section we recall some results concerning the norm of linear projections. Let Z be a uniformly convex Banach space, let V be a subspace of Z with codimension 1 and write

λ (V , Z) = inf {∥P∥ ∶ P is a linear projection of Z onto V } .

Any P at which λ (V , Z ) is attained is called a minimal projection (corresponding to V ).

As Z is uniformly convex, to each V there corresponds a unique minimal projection (see [10]). Now let

1 + α Z ∶= sup λ (V , Z) ,

where the supremum is taken over all subspaces V with codimension 1: evidently 1 ⩽ λ (V , Z ) ⩽ 1 + α Z ⩽ 2.

Following [11], we say that Z has the (α Z , 1) propery if given any subspace V of Z with codimension 1, there is a minimal projection P with ∥P∥ = 1 + α Z : in other words, λ (V , Z ) is independent of V . The following remarkable theorem is due to Rolewicz:

2.4. Theorem. Let p ∈ (1, ∞) and suppose that I is the unit interval (0, 1) ⊂ R. Then L ^p (I) has the (α p , 1) property, where α p = α L

(I) .

The value of α p was determined by Franchetti [5]:

2.5. Theorem. With the notation of Theorem 2.4,

1 + α p = max

m∈(0,1) (m ^p/p

′

+ (1 − m) ^p/p

′

)

1/ p

(m ^p

′

/ p

+ (1 − m) ^p

′

/ p

)

1/ p

. (5)

From (5) the estimate 1 + α p ⩽ 2 ^{2 p}

′

/ p

(obtained earlier by Rolewicz) follows easily when p > 2. Formula (5) also shows that, for example,

1 + α ₃ = 1 3

(17 + 7

√ 7)

1/3

≃ 1.0957, 1 + α ₄ = (1 + 2 3

√ 3)

1/4

≃ 1.21156.

The notion of a polar projection is useful. Given any uniformly convex, uniformly smo- oth real Banach space Z with duality map J Z , and any z ₁ ∈ Z with unit norm, let V z

∶=

{z ∈ Z ∶ ⟨z, J Z z ₁ ⟩ = 0} . The projection Q z

∶ Z → V z

defined by Q z

z = z − ⟨z, J Z z ₁ ⟩ z 1

is said to be a polar projection (of Z on V z

). Franchetti established the following theorem:

2.6. Theorem. With the notation of Theorem 2.4, every polar projection (of L p (I)) is a mi- nimal projection.

An immediate consequence of the above observations is that P ₂ ^Y is a polar projection, and if Y = L p (I), then ∥P ₂ ^Y ∥ = 1 + α p . As noted above, the restriction of P _n ^Y to Y _n−1 is a linear projection of Y _n−1 onto Y n . Taking

Q y

y ∶= y − ⟨ y, J Y y _n−1 ⟩ _Y y _n−1

we can see that the restriction of the polar projection Q y

to Y n−1 coincides with the restriction of P _n ^Y to Y _n−1 . Using this and Theorem 2.4 we have

sup

y∈Y

∥P _n ^Y ∥ _Y = sup

y∈Y

∥Q y

y∥ _Y ⩽ sup

y∈Y

∥Q y

y∥ _Y ⩽ (1 + α p ) ∥ y∥ _Y . We thus have

2.7. Lemma. Let n ∈ N ∖ {1}, x ∈ X, y ∈ Y , with X = Y = L ^p (I). Then

∥id − S ^Y _n ∥

Y →Y = ∥P n ^Y P _n−1 ^Y . . . P ₂ ^Y ∥

Y →Y ⩽ (1 + α p )

n−1

.

3. The main results

Let p ∈ (1, ∞), I = (0, 1), X = Y = L p (I) and suppose that T ∶ X → X is compact and linear, with trivial kernel and range dense in L p (I); denote the norm in X by ∥⋅∥ _p . We associate with T the objects x k , y k , X k , Y k , P _k ^Y , S k , λ k and α p introduced in the last section. Note that Y = L p (I).

3.1. Lemma. Suppose that λ k ⩽ (1 + α p ) ^{−k +1} for all k ∈ N. Then for all y ∈ T(B ^X ) and all n ∈ N,

∥ y − S _n ^Y y∥ _p ⩽ 1.

Proof. Let y ∈ T (B X ). By Lemma 2 of [4], Y = Y n ⊕ Z _n−1 ^Y , where Z ^Y _n−1 = sp { y 1 , . . . , y _n−1 } . If y ∈ Z ^Y _n−1 ,then P _n ^Y P _n−1 ^Y . . . P ₂ ^Y y = y − S ^Y _n y = 0. Thus

sup {∥P _n ^Y P _n−1 ^Y . . . P ₂ ^Y y∥ ∶ y ∈ T (B X )} = sup {∥P _n ^Y P _n−1 ^Y . . . P ₂ ^Y y∥ ∶ y ∈ Y n ∩ T (B X )}

⩽ (1 + α p ) ⁿ⁻¹ sup {∥ y∥ _p ∶ y ∈ Y n ∩ T (B X )}

⩽ (1 + α p ) ⁿ⁻¹ λ n ⩽ 1, and the Lemma follows.

3.2. Lemma. If c n (T ) ⩽ (2 ⁿ − 1)

−1

(1 + α p ) ⁻ⁿ⁺¹ for all n ∈ N, then ∥y − S ^Y n y∥ _p ⩽ 1 for all y ∈ T (B X ) and all n ∈ N.

Proof. Immediate from Lemma 3.1 and (2).

The main result of the paper now follows exactly as does Theorem 11 of [4].

3.3. Theorem. If c n (T ) ⩽ (2 ⁿ − 1) ⁻¹ (1 + α p ) ⁻ⁿ⁺¹ for all n ∈ N, then for all x ∈ X, T x = ∑ _n α ^Y _n (x ) y n ,

where α ^Y _n (⋅) = ξ ^Y _n (T ⋅) ∈ X ^∗ .

References

[1] D. E. Edmunds and W. D. Evans, Representation of compact linear operators in Banach spaces, Birkhäuser, Basel 2013.

[2] D. E. Edmunds and J. Lang, The j-eigenfunctions and s-numbers, Math. Nachr. 283 (2010), 463–477, DOI 10.1002/mana.200910221.

[3] D. E. Edmunds and J. Lang, Eigenvalues, embeddings and generalised trigonometric functions, Springer, Berlin 2011, DOI 10.1007/978-3-642-18429-1.

[4] D. E. Edmunds and J. Lang, Explicit representation of compact linear operators in Banach spaces via polar sets, Studia Math. 214 (2013), 265–278, DOI 10.4064/sm214-3-5.

[5] C. Franchetti, The norm of the minimal projection onto hyperplanes in L

[0, 1] and the radial constant, Boll.

U. M. I. (7) 4-B (1990), 803–821.

[6] O. Hanner, On the uniform convexity of L

and l

, Arkiv Mat. 3 (1956), 239–244.

[7] R. C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265–292.

[8] J. Lindenstrauss, On the modulus of smoothness and divergent series in Banach spaces, Mich. Math. J. 10 (1963), 241–252.

[9] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik

und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Berlin-New York 1979.

[10] V. P. Odinec, On a property of reflexive Banach spaces with a transitive norm, Bull. Acad. Pol. Math. 25 (1982), 353–357.

[11] S. Rolewicz, On minimal projections of the space L

[0, 1] on 1-codimensional subspace, Bull. Pol. Acad. Soc.

Math. 34 (1986), 151–153.

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