Extensions of linear operators from hyperplanes and strong uniqueness of best approximation in L(X,W)

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ScienceDirect Journal of Approximation Theory 246 (2019) 28–42 www.elsevier.com/locate/jat

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Extensions of linear operators from hyperplanes and strong uniqueness of best approximation in L(X, W ) Paweł Wójcik Institute of Mathematics, Pedagogical University of Cracow, Podchora¸z˙ych 2, 30-084 Kraków, Poland Received 20 June 2018; received in revised form 16 April 2019; accepted 19 May 2019 Available online 27 May 2019 Communicated by Paul Nevai

Abstract The aim of this paper is to present some results concerning the problem of minimal projections and extensions. Let X be a reflexive Banach space and let Y be a closed subspace of X of codimension one. Let W be a finite-dimensional Banach space. We present a new sufficient condition under which any minimal extension of an operator A ∈ L(Y, W ) is strongly unique. In this paper we show (in some circumstances) that if 1 < λ(Y, X ), then a minimal projection from X onto Y is a strongly unique minimal projection. Moreover, we introduce and study a new geometric property of normed spaces. In this paper we also present a result concerning the strong unicity of best approximation. c 2019 Elsevier Inc. All rights reserved. ⃝ MSC: 46B20; 41A35; 41A52; 41A65; 47A58 Keywords: Linear operators; Minimal extension; Minimal projection; Strongly unique minimal extension; Strongly unique best approximation

1. Introduction Throughout this paper we will work with Banach spaces over R. For reflexive Banach spaces X and W , L(X, W ) denotes the space of all bounded linear operators from X into W . When X = W we will write L(X ) for L(X, X ). B X and S X stand for the closed unit ball and the unit sphere in X , respectively. For a given A ∈ L(X, W ), we write M A := {x ∈ S X : ∥Ax∥ = ∥A∥}. If A is a compact operator, then M A ̸= ∅. The Birkhoff orthogonality in X is defined by: x⊥B y :⇔ ∀λ∈R : ∥x∥ ≤ ∥x + λy∥ (see [2,6] or [7]). For subspaces M and K in X we write E-mail address: [email protected]. https://doi.org/10.1016/j.jat.2019.05.003 c 2019 Elsevier Inc. All rights reserved. 0021-9045/⃝

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M⊥B K if m⊥B k for all m ∈ M and k ∈ K . For x ∈ X \{0} let J (x) denote the set of supporting functionals: J (x) := { f ∈ X ∗ : ∥ f ∥ = 1, f (x) = ∥x∥}. By the Hahn–Banach Theorem, the set J (x) is nonempty. A well known characterization of the Birkhoff orthogonality is due to James (see [7]): x⊥B y

⇔

∃ f ∈J (x) : f (y) = 0.

ρ+′ (x,

(1.1)

limt→0+ ∥x+t y∥−∥x∥ t

is called the norm derivative. Basic The mapping y) := ∥x∥ · information about the norm derivative one can find in [1,4]. When the norm comes from an inner product ⟨·|·⟩, we obtain ρ+′ (·, ·) = ⟨·|·⟩. The concept of the function ρ+′ is a generalization of an inner product. There are many examples of normed spaces that are not inner product spaces, so that the generalization is quite useful. By ExtD we will mean the set of extreme points of D. In [4] it was shown that ρ+′ (x, y)

= ∥x∥ · sup{ f (y) : f ∈ J (x)} = ∥x∥ · sup{ f (y) : f ∈ J (x) ∩ ExtB X ∗ }.

(1.2)

Now, suppose Y ⊆ X is a closed subspace of a Banach space X . An element yo ∈ Y is called a best approximation to x ∈ X iff ∥x − yo ∥ = dist(x, Y ) := inf{∥x − y∥ : y ∈ Y }. The set of all best approximations to x ∈ X in Y will be denoted by P Y (x) := {yo ∈ Y : ∥x − yo ∥ = dist(x, Y )}. It is worth mentioning that yo ∈ P Y (x)

⇔

x − yo ⊥B Y.

Let us recall that an element yo ∈ Y is called a strong unique best approximation (briefly, SUBA) for x ∈ X if and only if there exists r > 0 such that for every y ∈ Y , r · ∥y − yo ∥ + ∥x − yo ∥ ≤ ∥x − y∥.

(1.3)

It is obvious that if yo is the SUBA to x in Y then cardP Y (x) = 1, i.e. P Y (x) = {yo }. The converse is not true. Moreover, one should be able to verify with little effort that the following result holds. Theorem 1.1.

The following conditions are equivalent:

(i) yo is the SUBA to x in Y with constant r (yo ∈ P Y (x) r esp.), (ii) r · ∥x − yo ∥ · ∥y∥ ≤ ρ+′ (x − yo , y) for all y in Y \ {0} (0 ≤ inf{ρ+′ (x − yo , y) : y in Y \ {0}} resp.). { ( ) } x−yo y The constant r = inf ρ+′ ∥x−y , : y ∈ Y \ {0} is the best possible. o ∥ ∥y∥ Let Y be a finite-dimensional subspace. Then the closed unit ball BY is compact. So, on account of Theorem 1.1, since the function ρ+′ (x − yo , ·) is continuous (see [1,4]), as a corollary we get the following result. Theorem 1.2. Let X be a Banach space. If Y ⊆ X is a finite-dimensional subspace, then the following conditions are equivalent: (i) yo is the SUBA to x in Y , (ii) 0 < ρ+′ (x − yo , y) for all y in Y \ {0}.

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We will need the following result (cf. [12, Theorem 2.4], [13, Theorem 3.2]): Theorem 1.3 ([12,13]). Suppose that X is a reflexive Banach space. Let W be a normed space. Let A, T ∈ L(X, W ). If both operators A, T are compact, then the following conditions are true: { } (a) ρ+′ (A, T ) = sup {ρ+′ (Ax, T x) : x ∈ M A , } (b) ρ+′ (A, T ) = sup ρ+′ (Ax, T x) : x ∈ M A ∩ ExtB X . As a consequence of the above results and by properties of ρ+′ , one has as even more general result. Theorem 1.4. Let X, W, A, T be as in Theorem 1.3. Assume that N ⊆ S X . If (M A ∩ExtB X ) ⊆ (N ∪ −N ) ⊆ M A , then { } ρ+′ (A, T ) = ∥A∥ · sup w∗ (T x) : x ∈ N , w∗ ∈ ExtBW ∗ , w∗ (Ax) = ∥A∥ . (1.4) Proof. We use a simple fact that ρ+′ (♢, ♦) = ρ+′ (−♢, −♦). Now by applying Theorem 1.3 we { } (1.2) obtain ρ+′ (A, T ) = sup ρ+′ (Ax, T x) : x ∈ N = sup {sup{w∗ (T x) : w∗ ∈ ExtBW ∗ , w∗ (Ax) = ∥A∥} : x ∈ N } and we have our assertion. □ The paper is organized as follows. In Section 2 we state the main results of this note. In Section 3 we derive a sufficient condition under which any minimal extension of A is strongly unique. Moreover, we introduce and study a new geometric property of normed spaces. Another result of Section 3 says that if X is a four-dimensional normed space, Y := Z ⊕1 R ⊆ X is one of its hyperplanes and A ∈ L(Y ) is so chosen that there is no extension of A onto X which preserves the norm of A, then A has exactly one extension Ao of minimal norm. Furthermore, this extension is strongly unique. In Section 4 we mainly treat the case of spaces X = L(U, Z ) and W = HO(U, Z ) (the space of all homogeneous, continuous mappings going from U into W ). 2. Main result The aim of this section is to generalize [8, Theorem 1.3] from the finite dimensional case to the case of reflexive spaces. Now, suppose Y ⊆ X is a closed subspace of a Banach space X . We can consider a space LY (X, W ) := {T ∈ L(X, W ) : T |Y = 0}. We characterize strong unicity in terms of extreme points of the closed unit ball in W ∗ . Theorem 2.1 will be more convenient for applications. We are in position to prove the main result of this paper. Theorem 2.1. Assume that X is a reflexive Banach space and let f ∈ X ∗ , ∥ f ∥ = 1, Y = ker f . Suppose that W is an n-dimensional normed space. Let A ∈ L(X, W ), To ∈ LY (X, W ). Suppose that Ao := A − To does not attain its norm in Y (i.e., Y ∩ M Ao = ∅). Assume that ExtBW ∗ is closed. Put { } f (x) ∈ (0, +∞), ∗ ∗ FAo := w ∈ ExtBW : ∃x∈M Ao ∩ExtB X ∗ . w (Ao x) = ∥Ao ∥ Then the following conditions are true: (a) To ∈ P LY (X,W ) (A) if and only if 0 ∈ convFAo ; (b) To is a SUBA to A in LY (X, W ) if and only if 0 ∈ int convFAo . Moreover, the set FAo is closed.

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Before we begin to prove Theorem 2.1, a few words are appropriate. It seems that the set of extreme points of the closed unit ball would have to be closed. This is untrue even if the normed space is finite dimensional, as the following remark illustrates. For this reason we assume that ExtBW ∗ is closed. Consider the set K := {(x, y, 0) : x 2 + y 2 = 1} ∪ {(a, 0, b) : a, b ∈ {−1, 1}}. Let W := (R3 , ∥ · ∥ K ) be a three-dimensional normed space for which BW = convK . A moment’s reflection shows that ExtBW = K \ {(−1, 0, 0), (1, 0, 0)}. We want to point out when the assumption M A−To ∩Y = ∅ is fulfilled. It is worth mentioning that in many cases of hyperplanes and operators, M A−To ∩ Y = ∅. Indeed, let X be a finite dimensional Banach space. It is well known that if X is a real normed space, where the norm does not come from an inner product, then there exist a hyperplane Y in X and a linear projection P : X → Y satisfying ∥P∥ > 1. It is easy to check that M P ∩ Y = ∅. Now, we are going to prove Theorem 2.1. Proof. First we show that the set FAo is closed. Fix wo∗ ∈ clFAo . Suppose that wn∗ → wo∗ and wn∗ ∈ FAo . Thus wn∗ ∈ ExtBW ∗ . Since ExtBW ∗ is closed, we have wo∗ ∈ ExtBW ∗ . Since wn∗ ∈ FAo , it follows that f (xn ) ∈ (0, +∞) and wn∗ (Ao xn ) = ∥Ao ∥ for some xn ∈ M Ao ∩ExtB X . Recall that the space X is reflexive. Since B X is weak-compact, we may find a subsequence weak (x(n k ) such that xn k −→( xo )for some xo ∈ B X . In particular, ϕ(·) := wo∗ (Ao (·)) ∈ X ∗ and ) ϕ xn k → ϕ (xo ) and f xn k → f (xo ) ∈ [0, +∞). For each n k | ϕ(xo ) − ∥Ao ∥ | = |ϕ(xo ) − wn∗k (Ao xn k )| ≤ |ϕ(xo ) − wo∗ (Ao xn k )| + |wo∗ (Ao xn k ) − wn∗k (Ao xn k )|  ⏐ ⏐    ≤ ⏐ ϕ(xo ) − ϕ(xn k ) ⏐ + wo∗ − wn∗k  · ∥Ao ∥ · ∥xn k ∥   ⏐ ⏐   ≤ ⏐ ϕ(xo ) − ϕ(xn k ) ⏐ + wo∗ − wn∗k  · ∥Ao ∥ · 1 which tends to 0 as n k → ∞. This forces ϕ(xo ) = ∥Ao ∥. It is easy to check that ∥ϕ∥ = ∥Ao ∥. Thus the functional ϕ attains its norm at the point xo ∈ S X and xo ∈ M Ao . Since M Ao ∩ Y = ∅, f (xo ) ∈ (0, +∞). Moreover, the value ∥ϕ∥ is attained at some point xe ∈ ExtB X and consequently ∥Ao ∥ = ∥ϕ∥ = ϕ(xe ) = wo∗ (Ao (xe )). Therefore xe ∈ M Ao . It follows from the assumption (i.e. Y ∩ M Ao = ∅), that f (xe ) ̸= 0. An easy computation shows that we may assume f (xe ) ∈ (0, +∞). Hence wo∗ ∈ FAo , and we conclude that FAo is closed. Since the proofs are similar we present only one. Namely, we prove (b). In order to prove “⇐”, assume first that 0 ∈ int convFAo . We claim that ∀w∈W \{0} ∃w∗ ∈FAo w ∗ (w) > 0.

(2.1)

Suppose, for a contradiction, that ∃wa ∈W \{0} ∀w∗ ∈FAo w∗ (wa ) ≤ 0. It (follows from the above that FAo ⊆ {v ∗ ∈ W ∗ : v ∗ (wa ) ≤ 0}. Therefore, we obtain ) int convFAo ⊆ int (conv {v ∗ ∈ W ∗ : v ∗ (wa ) ≤ 0}). By assumption ( ) ( { }) 0 ∈ int convFAo ⊆ int conv v ∗ ∈ W ∗ : v ∗ (wa ) ≤ 0 { ∗ } { } = int v ∈ W ∗ : v ∗ (wa ) ≤ 0 = v ∗ ∈ W ∗ : v ∗ (wa ) < 0 , and we obtain a contradiction. Thus the condition (2.1) holds.

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Next we prove that To is a SUBA to A in LY (X, W ). Fix T ∈ LY (X, W ) \ {0}. Since ker f = Y , we have T (x) = f (x) · wo for some wo ∈ W \ {0}. It follows from (2.1) that there exists wo∗ in FAo such that wo∗ (wo ) > 0. Since wo∗ ∈ FAo , there exists xo ∈ M Ao ∩ ExtB X such that wo∗ (Ao xo ) = ∥Ao ∥ and f (xo ) ∈ (0, +∞). Moreover, dim LY (X, W ) = dim W < ∞. Using Theorem 1.3, we obtain (1.2)

ρ+′ (A − To , T ) = ρ+′ (Ao , T ) ≥ ρ+′ (Ao xo , T xo ) ≥ wo∗ (T xo ) = wo∗ ( f (xo ) · wo ) = f (xo ) · wo∗ (xo ) > 0, and from Theorem 1.2 we deduce that To is a SUBA to A in LY (X, W ). In order to prove “⇒”, suppose that To is a SUBA to A in LY (X, W ). Assume, for a contradiction, that 0 ∈ / int convFAo . Then there is a closed hyperplane H ⊆ W ∗ separating }int convFAo { ∗ ∗ from {0 }; that is, there is a wo in W \{0} such that sup w (wo ) : w ∗ ∈ int convFAo ≤ 0∗ (wo ). In particular, { } sup w∗ (wo ) : w∗ ∈ FAo ≤ 0. (2.2) ˆ ∈ L(X, W ) defined as follows We consider the operator T ˆ(x) := f (x) · wo , T

x ∈ X.

Using Theorem 1.4 we have ⎫ ⎧ ⎪ ⎪ x ∈ M ∩ ExtB , ⎪ ⎪ A X o ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ ( ) f (x) ∈ (0, +∞), (1.4) ∗ ′ ′ ˆx : ˆ) = ρ+ (Ao , T ˆ) = ∥Ao ∥ · sup w T ρ+ (A − To , T ⎪ ⎪ w ∗ ∈ ExtBW ∗ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ∗ w (Ao x) = ∥Ao ∥ ⎫ ⎧ ⎪ x ∈ M Ao ∩ ExtB X ,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ f (x) ∈ (0, +∞), ∗ = ∥Ao ∥ · sup w ( f (x) · wo ) : ⎪ ⎪ w∗ ∈ ExtBW ∗ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ∗ w (Ao x) = ∥Ao ∥ ⎧ ⎫ ⎪ x ∈ M Ao ∩ ExtB X ,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ (2.2) f (x) ∈ (0, +∞), ∗ = ∥Ao ∥ · sup f (x) · w (wo ) : ≤ 0. ⎪ ⎪ w∗ ∈ ExtBW ∗ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ w∗ (Ao x) = ∥Ao ∥ ˆ) ≤ 0. Moreover T ˆ ∈ LY (X, W ) \ {0}. This is in From this it follows that ρ+′ (A − To , T contradiction to the conclusion of Theorem 1.2, and we are done. In order to prove (a), it is necessary to show that FAo is closed. But we had proved that it is true. The rest of the proof is similar to that of (b). □ It is interesting to look at FAo in some special cases. Indeed, the next theorem will be used in Section 3.3 to present a result concerning strong unicity. Theorem 2.2.

Let X, Y, W, A, To be as in Theorem 2.1. Then FAo ∩ −FAo = ∅.

Proof. Suppose, for a contradiction, that w∗ ∈ FAo ∩ −FAo for some w∗ ∈ W ∗ . It follows that w ∗ , −w ∗ ∈ FAo . Thus there are x1 , x2 in M Ao such that f (x1 ), f (x2 ) ∈ (0, +∞) and

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w ∗ (Ao x1 ) = ∥Ao ∥, −w∗ (Ao x2 ) = ∥Ao ∥. Fix a number λ ∈ [0, 1]. It follows that ( ) ∥Ao ∥ = λ∥Ao ∥ + (1 − λ)∥Ao ∥ = λw ∗ (Ao x1 ) + (1 − λ) −w ∗ (Ao x2 ) ≤ w ∗ (Ao (λx1 + (1 − λ)(−x2 ))) ≤ ∥Ao (λx1 + (1 − λ)(−x2 ))∥ ≤ ∥Ao ∥ · ∥λx1 + (1 − λ)(−x2 )∥ ≤ ∥Ao ∥. Therefore ∥Ao (λx1 + (1 − λ)(−x2 ))∥ = ∥Ao ∥. Thus λx1 + (1 − λ)(−x2 ) ∈ M Ao . From this it is immediate to infer that conv{x1 , −x2 } ⊆ M Ao . On the other hand, f (−x2 ) < 0 < f (x1 ) so there is a number λa ∈ (0, 1) such that f (λa x1 + (1 − λa )(−x2 )) = 0. But then M Ao ∩ ker f ̸= ∅, i.e. M Ao ∩ Y ̸= ∅, a contradiction. □ We are going to present some new applications of the above theorem, which generalize some results from [8,9]. However, our result holds more generally. 3. On codimension-one minimal extensions 3.1. Preliminaries Let W be real Banach spaces. Suppose that Y ⊆ X is a closed subspace of a Banach space X . Given A ∈ L(Y, W ), we write P A (X, W ) := {E ∈ L(X, W ) : E|Y = A}, so that P A (X, W ) is the set of all extensions of A. We say that E o ∈ P A (X, W ) is minimal if ∥E o ∥ = λ A (Y, X ) := inf{∥E∥ : E ∈ P A (X, W )}. In particular, if A = I d ∈ L(Y ), the set P(X, Y ) := P I d (X, Y ) contains all linear, continuous projections from X onto Y . Obviously, if {0} ̸= Y ⊊ X , then 1 ≤ ∥P∥ for any projection P ∈ P(X, Y ). It is easy to check that E o is minimal if and only if 0 ∈ P LY (X,W ) (E o ). For E o with a minimal norm in P A (X, W ), the condition (1.3) becomes r · ∥E − E o ∥ + ∥E o ∥ ≤ ∥E∥ for all E ∈ P A (X, W ). Then we say that E o is strongly unique minimal extension of A (briefly, SUM-extension). It is not difficult to show that E o is SUM-extension if and only if 0 is SUBA for E o in LY (X, W ). The main tool in our approach in this section is Theorem 2.1 which characterizes the SUM-extensions of operators. 3.2. On the geometry of dual spaces Recall that an n-dimensional subspace H of C[a, b] is called a Haar subspace if and only if for any set {t1 , . . . , tn } of distinct points from [a, b], 0 is the only element in H which vanishes on {t1 , . . . , tn }. In this paper we will consider the following property which is essentially weaker than Haar spaces. Namely, we say that an n-dimensional space X has property (IE) if: ⎛ ∗ ⎞ ( ) x1 , . . . , xk∗ ∈ ExtB X ∗ , k ≤ n ∗ ∗ ⎝xi∗ ̸= x ∗j for i ̸= j ⎠ H⇒ the set {x1 , . . . , xk } . (3.1) is linearly independent −x ∗j ∈ / {x1∗ , . . . , xk∗ } for j ≤ k We can think of (IE) as “Independent Extreme points”. Obviously, any Haar space H satisfies (IE). Indeed, fix x1∗ , . . . , xn∗ ∈ ExtB H ∗ . For t ∈ [a, b], let ηt : C[a, b] → R denote the evaluation function on C[a, b] defined by ηt (g) := g(t). We have ExtB H ∗ ⊆ {±ηt : t ∈ [a, b]} (see e.g. [3, Corollary 3.4]). Therefore may assume, without loss of generality, that x1∗ = ηt1 , . . . , xn∗ = ηtn and ti ̸= t j for i ̸= j. Let { f 1 , . . . , f n } ⊆ H be a basis for H . Since H is a Haar space, the map H ∋ f ↦→ ( f (t1 ), . . . , f (tn )) ∈ Rn is invertible iff the map

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(∑n ) ∑n Rn ∋ c ↦ → i=1 ci f i (t1 ), . . . , i=1 ci f i (tn ) is invertible. From this it follows that the set {x1∗ , . . . , xn∗ } is linearly independent. To summarize, we have shown that Haar spaces satisfy (IE). On the other hand, it is easy to see that the space Y := span{ pa (t) := 1 − t 2 , pb (t) := t − t 3 , pc (t) := t 2 − t 4 } ⊆ C[−1, 1], which is not Haar space, satisfies (IE). We omit the proof — a more general result will be proved later (Theorem 3.5). Actually, one can consider Y = (R3 , ∥ · ∥) with the norm for which the unit ball is a symmetric convex polyhedron D ⊆ Y such that card(M ∩ ExtD) ≤ 4 for any two-dimensional linear subspace M ⊆ Y (in particular, ExtD = the vertices of the polyhedron D). Then the condition (3.1) is satisfied for the space X := Y ∗ . This means that X has property (IE). The property (IE) plays a crucial role in the next subsection. To illustrate some of the nuances of this property, we give some examples of such polyhedrons. The regular polyhedron with 12 faces (or the regular polyhedron with 20 faces) can serve as an example. The convex polyhedron with 12 congruent rhombic faces is also a suitable example. Another interesting result is given in: Proposition 3.1.

n The space l∞ has property (IE).

The idea of the proof is rather simple, so we omit the proof. Note that the cube and the regular polyhedron with 8 faces are special cases of Proposition 3.1 and Corollary 3.4 (with n = 3). The following theorem gives a great number of examples of spaces satisfying (IE). Theorem 3.2. satisfies (IE).

Let Y be a two-dimensional normed space. Then the space X := Y ⊕1 R

Proof. It is easy to see that dim X = 3 and X ∗ = Y ∗ ⊕∞ R. We show that X satisfies (3.1). For k = 1 or k = 2 the assertion is obvious. So we prove only the case k = 3. Fix x1∗ , x2∗ , x3∗ ∈ ExtB X ∗ and assume xi∗ ̸= x ∗j for i ̸= j. Suppose that −x ∗j ∈ / {x1∗ , x2∗ , x3∗ } for j ∈ {1, 2, 3}. Suppose, for a contradiction, that the set {x1∗ , x2∗ , x3∗ } is not linearly independent. Without loss of generality, we may assume that λ1 x1∗ + λ2 x2∗ = x3∗ , λ1 ̸= 0. It is easy to check that x ∗j = (y ∗j , ±1) for some y ∗j ∈ ExtBY ∗ . We may assume without loss of generality that x1∗ = (y1∗ , 1), x2∗ = (y2∗ , 1), x3∗ = (y3∗ , 1). This clearly forces yi∗ ̸= y ∗j for i ̸= j and (λ1 y1∗ + λ2 y2∗ , λ1 + λ2 ) = λ1 (y1∗ , 1) + λ2 (y2∗ , 1) = λ1 x1∗ + λ2 x2∗ = x3∗ = (y3∗ , 1). It follows from the above equalities that λ1 y1∗ + λ2 y2∗ = y3∗

and

λ1 + λ2 = 1.

(3.2)

This leads to three cases: λ1 , λ2 ∈ [0, 1] or λ1 ∈ / [0, 1] or λ2 ∈ / [0, 1]. In the first case, we obtain y3∗ ∈ conv{y1∗ , y2∗ }. Thus y3∗ ∈ / ExtBY ∗ or y3∗ = y1∗ or y3∗ = y2∗ . This is a contradiction. Considering the other cases (i.e. λ1 ∈ / [0, 1] or λ2 ∈ / [0, 1]), and recalling that y1∗ , y2∗ ∈ ∗ ExtBY ∗ , the above computation (3.2) says y3 ∈ / BY ∗ . This is, again, a contradiction, and we are done. □ Unfortunately, the analogous statement does not hold more generally. Theorem 3.3. Let Y be a normed space such that 3 ≤ dim Y = n. Then the space X := Y ⊕1 R does not satisfy (IE).

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Proof. It is easily seen that dim X ≥ 4 and X ∗ = Y ∗ ⊕∞ R. Fix (y1∗ , 1), (−y1∗ , 1), (y2∗ , 1), (−y2∗ , 1) ∈ ExtB X ∗ and assume y1∗ ̸= y2∗ ̸= −y1∗ . It follows that 1 · (y1∗ , 1) + 1 · (−y1∗ , 1) + (−1) · (y2∗ , 1) = (−y2∗ , 1). Thus the functionals (y1∗ , 1), (−y1∗ , 1), (y2∗ , 1), (−y2∗ , 1) do not satisfy (3.1). The proof of Theorem 3.3 is complete. □ Corollary 3.4.

The space l1n has property (IE) iff n < 4.

The next result yields a large number of examples of normed spaces satisfying (3.1), i.e. (IE). Theorem 3.5. Let Ω ⊆ R be a compact subset. Let h, f ∈ C(Ω ). Assume that f is injective. Define N (h) { := {t ∈ Ω : h(t) = } 0} and suppose that ∅ ̸= N (h) ̸= Ω . Then the space X := span h, h f, h f 2 , . . . , h f m−1 has property (IE). Moreover, if Ω = [a, b], then X is not a Haar subspace. Proof. It is easy to see that dim X =: n ≤ m. Fix k ∈ {1, 2, . . . , n}. Let x1∗ , . . . , xk∗ ∈ ExtB X ∗ . Suppose that the functionals x1∗ , . . . , xk∗ are distinct. Assume that −x ∗j ∈ / {x1∗ , . . . , xk∗ } for j ∈ {1, . . . , k}. For t ∈ Ω , let ηt : Ω → R denote the evaluation function on C(Ω ) defined by ηt (g) := g(t). It follows from (cf. [3, Corollary 3.4]) that ExtB X ∗ ⊆ {±ηt : t ∈ Ω }. So we may assume, without loss of generality, that x1∗ = ηt1 , . . . , xk∗ = ηtk and ti ̸= t j for i ̸= j. It is easily seen that if t ∈ N (h), then ηt = 0 on X . Since the functionals ηt1 , . . . , ηtk ∈ ExtB X ∗ are nonzero, we get t j ∈ / N (h) for j ∈ {1, . . . , k}. It suffices to show that ηt1 , . . . , ηtk are linearly independent. Suppose that α1 ηt1 + α2 ηt2 + . . . + αk ηtk = 0. Thus α1 . . . , αk satisfy the system of linear equations ⎧ + α2 h(t2 ) + . . . + αk h(tk ) = 0, ⎪ ⎪α1 h(t1 ) ⎪ ⎪ + α2 h(t2 ) f (t2 ) + . . . + αk h(tk ) f (tk ) = 0, ⎨α1 h(t1 ) f (t1 ) α1 h(t1 ) f 2 (t1 ) + α2 h(t2 ) f 2 (t2 ) + . . . + αk h(tk ) f 2 (tk ) = 0, ⎪ ⎪ . . . . . . . . . . . . . . . . . . . . . . . . ... ⎪ ⎪ ⎩ α1 h(t1 ) f k−1 (t1 ) + α2 h(t2 ) f k−1 (t2 ) + . . . + αk h(tk ) f k−1 (tk ) = 0. The determinant of this system is ⎡ ⎤ h(t1 ) h(t2 ) . . . h(tk ) ⎢h(t1 ) f (t1 ) h(t2 ) f (t2 ) . . . h(tk ) f (tk ) ⎥ ⎢ ⎥ 2 2 ⎢ h(t2 ) f (t2 ) . . . h(tk ) f 2 (tk ) ⎥ det ⎢h(t1 ) f (t1 ) ⎥ ⎣. . . . . . ⎦ ...... ... ...... k−1 k−1 k−1 (t1 ) h(t2 ) f (t2 ) . . . h(tk ) f (tk ) h(t1 ) f ⎤ ⎡ 1 1 ... 1 ⎢ f (t1 ) f (t2 ) . . . f (tk ) ⎥ ⎥ ⎢ 2 2 f (t ) f (t ) . . . f 2 (tk ) ⎥ = h(t1 )h(t2 ) . . . h(tk ) · det ⎢ 1 2 ⎢ ⎥ ⎣. . . . . . ...... ... ...... ⎦ f k−1 (t1 ) f k−1 (t2 ) . . . f k−1 (tk ) ∏ = h(t1 )h(t2 ) . . . h(tk ) · ( f (t j ) − f (ti )). 1≤i< j≤k

The above equality follows from the well known facts about Vandermonde matrices. But f is ∏ injective, so 1≤i< j≤k ( f (t j ) − f (ti )) ̸= 0. Since the associated determinant is different than zero, we have α1 = · · · = αk = 0. This means that ηt1 , . . . , ηtk are linearly independent. Now suppose that Ω = [a, b]. We will show that X ⊆ C[a, b] is not a Haar subspace. Assume, for a contradiction, that X is a Haar subspace. Then there exists x ∈ X such that

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x(t) > 0 for all t ∈ [a, b] (see e.g. [10, Lemma 5, p. 201]). But if t ∈ N (h), then x(t) = 0. This is a contradiction. This completes the proof. □ The following result shows that the property (IE) is inherited by some subspaces. Theorem 3.6. Assume that X is an n-dimensional Banach space and let W ⊆ X be a closed hyperplane such that W = ker f and f ∈ ExtB X ∗ . Suppose that X satisfies (IE). Then W satisfies (IE). Proof. Fix w1∗ , . . . , wh∗ in ExtBW ∗ with h ≤ n − 1 = dim W . Suppose that wi∗ ̸= w ∗j for i ̸= j and −w∗j ∈ / {w1∗ , . . . , wh∗ } for j ≤ h. Suppose, for a contradiction, that {w1∗ , . . . , wh∗ } is a linearly dependent set in W ∗ . There exist h numbers λ1 , . . . , λh ∈ R such that h ∑

λk wk∗ = 0

and λ jo ̸= 0 ̸= λio for some jo , i o ∈ {1, . . . , h}, jo ̸= i o .

(3.3)

k=1

It follows from [3, p. 257] (see also [11] or [5, p.33,44]) that an extreme point of ExtBW ∗ must be restriction to W of an extreme point of ExtB X ∗ . Therefore for all k ∈ {1, . . . , h} there exists w ˆk∗ ∈ ExtB X ∗ such that w ˆk∗ |W = wk∗ . We see at once that −ˆ w∗j ∈ / {ˆ w1∗ , . . . , w ˆh∗ } for any j ≤ h. ∗ ∗ It is obvious that w ˆ j ̸= w ˆk for j ̸= k. This leads to two cases. ∑h Possibility 1. ˆk∗ = 0. Since X satisfies (IE), the set {ˆ w1∗ , . . . , w ˆh∗ } is linearly k=1 λk w independent. We thus get λ1 = . . . = λh = 0. This is a contradiction to the second part of (3.3). So we may exclude Possibility 1. ∑h ∑h Possibility 2. k=1 λk w ˆk∗ ̸= 0. Let g ∈ X ∗ be a linear functional defined by g := k=1 λk w ˆk∗ . Applying (3.3) we get W = ker g. Since ker f = ker g, we have β f = g for some β ∈ R \ {0}. Thus h ∑ βf = λk w ˆk∗ . (3.4) k=1

It follows easily that the set { f, w ˆ1∗ , . . . , w ˆh∗ } is linearly dependent. Now we show that f ∈ {ˆ w1∗ , . . . , w ˆh∗ }

or

ˆh∗ }. f ∈ −{ˆ w1∗ , . . . , w {ˆ w1∗ , . . . , w ˆh∗ }

(3.5) −{ˆ w1∗ , . . . , w ˆh∗ }.

Suppose, for a contradiction, that f ∈ / ∪ It is helpful to recall that f, w ˆ1∗ , . . . , w ˆh∗ ∈ ExtB X ∗ . Bearing in mind property (IE) and an inequality h < dim X , we that { f, w ˆ1∗ , . . . , w ˆh∗ } is linearly independent. This is in contradiction to β f = ∑h conclude ∗ ˆk . Thus we have proved (3.5). k=1 λk w Since f ∈ {ˆ w1∗ , . . . , w ˆh∗ } ∪ −{ˆ w1∗ , . . . , w ˆh∗ }, we may assume, without loss of generality, that ∑h ∗ ∗ ∗ f =w ˆ1 or f = −ˆ w1 (briefly f = ±ˆ w1 ). From (3.4) we have 0 = (∓β + λ1 )ˆ w1∗ + k=2 λk w ˆk∗ . ∗ ∗ Since X satisfies (IE), it follows that the set {ˆ w1 , . . . , w ˆh } is linearly independent. Thus, in particular, λ2 = · · · = λh = 0. This gives λ jo = 0 or λio = 0. This is a contradiction to the inequalities λ jo ̸= 0 ̸= λio in (3.3). Thus we can exclude Possibility 2 and the proof of Theorem 3.6 is complete. □ As a corollary we get another result involving the property (IE). Theorem 3.7. Assume that X is an n-dimensional Banach space and let W ⊆ X be a closed hyperplane such that x⊥B W and x ∈ S X is a point of smoothness. If X satisfies (IE), then W satisfies (IE).

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Proof. Smoothness guarantees that there is a unique supporting functional (at x), denoted by f x , that is J (x) = { f x }. An easy computation shows that f x ∈ ExtB X ∗ . Furthermore, applying (1.1) we get W = ker f x , and on the account of Theorem 3.6 we have our assertion. □ To end this subsection we present yet another example. Consider the vectors a1 = (1, 1, 0), a2 = (1, −1, 0), a3 = (−1, −1, 0), a4 = (−1, 1, 0) ∈ R3 , and ( b1 =

) ( ) ( ) ( ) 1 1 1 1 , 0, 1 , b2 = − , 0, 1 , b1 = , 0, −1 , b1 = − , 0, −1 ∈ R3 . 2 2 2 2

Define D ⊆ R3 by D := conv{a1 , a2 , a3 , a4 , b1 , b2 , b3 , b4 }. Let Y := (R3 , ∥ · ∥) be a threedimensional normed space for which BY = D. Then the space X := Y ∗ has property (IE). Indeed, a moment’s reflection shows that ExtB X ∗ = {a1 , a2 , a3 , a4 , b1 , b2 , b3 , b4 } satisfies (3.1). 3.3. Extensions of linear operators from hyperplanes The aim of this subsection is to generalize [9, Theorem 6] from the Haar spaces case to the case of (IE). More precisely, let Hn ⊆ C[a, b] be an n-dimensional Haar subspace and Hn−1 be an n − 1-dimensional Haar subspace of Hn . Lewicki and Micek [9, Theorem 6] considered an operator A ∈ L(Hn−1 , Hn−1 ) and its extension E ∈ L(Hn , Hn−1 ), i.e. E| Hn−1 = A. Therefore, in the theorem below, we generalize the result from [9, Theorem 6] to the case of extensions of operator A ∈ L(Y, W ); that is, E ∈ L(X, W ) with E|Y = A. We show that if a norm of minimal extension of A from X into W is greater than the operator norm of A : Y → W , then it is a strongly unique minimal extension. Now we state the first main result of this subsection. Theorem 3.8. Assume that X is a reflexive Banach space and let Y ⊆ X be a closed hyperplane. Suppose that W is an n-dimensional normed space that satisfies (IE). Assume that ExtBW ∗ is closed. Assume furthermore A ∈ L(Y, W ), ∥A∥ < λ A (Y, X ). Then a minimal extension of A in a set P A (X, W ) is strongly unique. Proof. From the assumption, we have that Y = ker f for some f ∈ X ∗ , ∥ f ∥ = 1. Assume that E o ∈ P A (X, W ) is a minimal extension of A : Y → W . This forces 0 ∈ P LY (X,W ) (E o ). So, we may write E o instead of E o − 0. We define { } f (x) ∈ (0, +∞), ∗ FEo := w ∈ ExtBW ∗ : ∃x∈M Eo ∩ExtB X ∗ . w (E o x) = ∥E o ∥ From the assumption (i.e. ∥A∥ < λ A (Y, X )), we have that M Eo ∩ Y = ∅. Suppose, for a contradiction, that this minimal extension is not strongly unique. This means that 0 is not a SUBA to E o in LY (X, W ). Theorem 2.1 now yields 0 ∈ convFEo and 0∈ / int convFEo . From this we conclude that there exist h extreme points w1∗ , . . . , wh∗ ∈ ExtBW ∗ and h numbers λ1 , . . . , λh > 0 such that h ∑ k=1

λk = 1

and

h ∑

λk wk∗ = 0.

(3.6)

k=1

Since dim W = n, by Carath´eodory’s Theorem we can assume h ≤ n + 1. However, since 0 ∈ / int convFEo , we can assume h ≤ n. Without loss of generality, we may assume that w1∗ , . . . , wh∗ are distinct. By Theorem 2.2, −w∗j ∈ / {w1∗ , . . . , wh∗ } for any j ≤ h. Since W

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satisfies (IE), the set {w1∗ , . . . , wh∗ } is linearly independent (see (3.1)). Thus the second part of (3.6) implies that λ1 = · · · = λh = 0. This is a contradiction to the first part of (3.6). The proof of Theorem 3.8 is complete. □ As an immediate consequence of Theorems 3.2 and 3.8, we have the following result. Theorem 3.9. Suppose that Z is two-dimensional. Let X be a four-dimensional Banach space such that Z ⊕1 R ⊆ X . Assume furthermore A ∈ L(Z ⊕1 R), ∥A∥ < λ A (Z ⊕1 R, X ). Suppose that ExtB Z ∗ is closed. Then A has exactly one extension E o of minimal norm. Moreover, this extension is strongly unique. Another interesting result combining (IE), strong unicity and minimal projections is given in the following theorem. A projection Po ∈ P(X, Y ) is called a strongly unique minimal projection (briefly a SUM projection) if and only if there is r > 0 such that r ·∥P − Po ∥+∥Po ∥ ≤ ∥P∥ for any P ∈ P(X, Y ). Theorem 3.10. Assume that X is an n-dimensional Banach space and let Y ⊆ X be a hyperplane. Suppose that Y satisfies (IE). Let Po ∈ P(X, Y ) be a minimal projection such that 1 < λ(Y, X ) = ∥Po ∥. Assume that ExtBY ∗ is closed. Then Po is a SUM projection. Proof. We consider the case where A = Po ∈ P(X, Y ) is the minimal projection. So the result follows from Theorem 3.8. □ Our next theorem is an application of Theorem 3.8. More precisely, the following result is an easy consequence of Theorems 3.6 and 3.8. Theorem 3.11. Assume that X is an m-dimensional Banach space and let W ⊆ X be a closed hyperplane such that W = ker f and f ∈ ExtB X ∗ . Assume that ExtBW ∗ is closed. Suppose that X that satisfies (IE). Assume furthermore A ∈ L(W ), ∥A∥ < λ A (W, X ). Then a minimal extension of A in a set P A (X, W ) is strongly unique. To end this section we present yet another application of Theorem 3.8. Theorem 3.12. Suppose that X is a finite-dimensional Banach space and let f ∈ X ∗ , ∥ f ∥ = 1, Y = ker f . Assume that W is an n-dimensional smooth normed space such that card(ExtB X \Y ) < 2·dim W +2. Let A ∈ L(X, W ), To ∈ LY (X, W ). Suppose that Ao := A−To does not attain its norm in Y . If To ∈ P LY (X,W ) (A), then To is not a SUBA to A in LY (X, W ). We should point out that the following proof does not use the property (IE). Moreover, since W is smooth, W ∗ is strictly convex, whence ExtBW ∗ = SW ∗ . Thus ExtBW ∗ is closed. Proof. Suppose that To ∈ P LY (X,W ) (A). Assume, for a contradiction, that To is a SUBA to A in LY (X, W ). Applying Theorem 2.1 we get that 0 ∈ int convFAo . From this (and from Carath´eodory’s Theorem) we conclude that there exist n + 1 distinct extreme points ∑n+1 ∗ w1∗ , . . . , wn+1 ∈ FAo and n + 1 numbers λ1 , . . . , λn+1 > 0 such that k=1 λk = 1 and ∑n+1 ∗ ∗ ∗ ∗ k=1 λk wk = 0. Since w1 , . . . , wn+1 ∈ FAo , it follows that w1 (Ao x 1 ) = ∥Ao ∥, . . ., ∗ wn+1 (Ao xn+1 ) = ∥Ao ∥ for some x1 , . . . , xn+1 ∈ ExtB X with f (x1 ) > 0, . . ., f (xn+1 ) > 0. We define a number m := 21 · card(ExtB X \ Y ). Let us denote ExtB X \ Y = ExtB X \ ker f = {e1 , . . . , em } ∪ {−e1 , . . . , −em },

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where f (e1 ) > 0, . . . , f (em ) > 0. Since f (x1 ) > 0, . . . , f (xn+1 ) > 0, it follows from the above equalities that {x1 , . . . , xn+1 } ⊆ {e1 , . . . , em }.

(3.7)

The inequality card(ExtB X \Y ) < 2·dim W +2 yields m < n+1. Thus it follows from (3.7) that there are numbers i, j ∈ {1, . . . , n + 1} and k ∈ {1, . . . , m} such that i ̸= j and xi = x j = ek . It may be assumed without loss of generality that x1 = x2 = e1 . Therefore we can write w1∗ (Ao e1 ) = ∥Ao ∥ and w2∗ (Ao e1 ) = ∥Ao ∥. Thus w1∗ (Ao e1 ) = w2∗ (Ao e1 ) = ∥Ao e1 ∥; that is, ∗ w1∗ , w2∗ ∈ J (Ao e1 ). Since W is smooth, w1∗ = w2∗ . Recall that the functional w1∗ , . . . , wn+1 are distinct. This is a contradiction. □ By Theorem 3.12 we get immediately. Theorem 3.13. Suppose that X is a finite-dimensional Banach space and let Y ⊆ X be a hyperplane. Assume that W is an n-dimensional smooth normed space such that card(ExtB X \ Y ) < 2 · dim W + 2. Let A ∈ L(Y, W ), ∥A∥ < λ A (Y, X ). Then a minimal extension of A in the set P A (X, W ) := {E ∈ L(X, W ) : E|Y = A} is not strongly unique. Corollary 3.14. Let Y ⊆ l1m be a hyperplane in l1m . Suppose that m < n + 1, p ∈ (1, ∞). Let A ∈ L(Y, l np ). If ∥A∥ < λ A (Y, l1m ), then a minimal extension of A is not strongly unique. k k Corollary 3.15. Let Y ⊆ l∞ be a hyperplane in l∞ . Suppose that 2k−1 < n + 1, p ∈ (1, ∞). n k Let A ∈ L(Y, l p ). If ∥A∥ < λ A (Y, l∞ ), then a minimal extension of A is not strongly unique.

4. On some minimal projections onto L(U, Z) In this part of the article we present some applications of the main results from Section 3. Let U, Z be real Banach spaces. The Banach space of all continuous homogeneous mappings from U into Z is denoted by HO(U, Z ). This means that { } HO(U, Z ) := A : U → Z | A is continuous and ∀α∈R,x∈U A(αx) = α A(x) , and the norm is defined by ∥A∥ := sup{∥A(x)∥ : x ∈ BU }. It is worth noting that ∥A∥ < ∞ if dim U < ∞. It should be clear that L(U, Z ) ⊆ HO(U, Z ). Example. Suppose that U is an n-dimensional normed space with n ≥ 2. Let L ∈ L(U, Z ) and put ⎧ ⎨ ∥L x∥ · L x if x ∈ X \ {0}, F(x) := ∥x∥ ⎩ 0 if x = 0. Let ∥L∥ = 1 and assume that L is not a linear isometry. A trivial verification shows that F ∈ HO(U, Z ). Claim 4.1. F ∈ / L(U, Z ); that is, L(U, Z ) ⊊ HO(U, Z ). Suppose, for a contradiction, that F ∈ L(U, Z ). We consider two cases. Case 1. Suppose that L is injective. Since L is not an isometry, there are linear independent vectors x, y ∈ U such that ∥x∥ = ∥y∥ = 1 and ∥L(x)∥ < ∥L(y)∥. It is clear that

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y∥ x∥ F(x + y) = F(x) + F(y) and consequently ∥L(x+y)∥ L(x + y) = ∥L L x + ∥L L y. From ∥x+y∥ ∥x∥ ∥y∥ ( ) ∥L(x+y)∥ ∥L(x+y)∥ this we conclude L ∥x+y∥ x + ∥x+y∥ y = L (∥L x∥x + ∥L y∥y). Since L is injective, we

see that ∥L(x+y)∥ ∥x+y∥

∥L(x+y)∥ x ∥x+y∥

+ ∥L(x+y)∥ y = ∥L x∥x + ∥L y∥y. Since x, y are linear independent, we have ∥x+y∥ = ∥L x∥ and ∥L(x+y)∥ = ∥L y∥. We thus get ∥L(x)∥ = ∥L(y)∥, which is impossible. ∥x+y∥

Case 2. Suppose that L is not injective. Fix x ∈ ker L, y ∈ U \ ker L such that ∥x∥ = ∥y∥ = 1. Fix α ∈ (0, +∞). It is clear that F(x + αy) = F(x) + F(αy) and L(x + αy) = ∥L(x)∥ L(x) + ∥L(αy)∥ L(αy). From x ∈ ker L we deduce consequently ∥L(x+αy)∥ ∥x+αy∥ ∥x∥ ∥αy∥ ∥L(αy)∥ ∥L y∥ y∥ ∥L(x+αy)∥ L(αy) = ∥αy∥ L(αy) = ∥y∥ L(αy). Since L(αy) ̸= 0, we get ∥L(x+αy)∥ = ∥L . Since ∥x+αy∥ ∥x+αy∥ ∥y∥ y∥ ∥L x∥ α was arbitrarily chosen from the interval (0, +∞), letting α → 0+ we obtain ∥x∥ = ∥L . ∥y∥ On the other hand, we have ∥L x∥ = 0 ̸= ∥L y∥ and ∥x∥ = ∥y∥ = 1. This is a contradiction. Let X be a subspace such that L(U, Z ) ⊊ X ⊆ HO(U, Z ). In this section we show that a minimal projection Po : X → L(U, Z ) satisfies the inequality 1 < ∥Po ∥. Moreover, we will apply Theorem 3.10 in the case when L(U, R) ⊊ X ⊆ HO(U, R). In other words, we will examine when a projection Po : X → L(U, R) is a strongly unique minimal projection in P (X, L(U, R)), i.e. when Po : X → L(U, R) satisfies 1 < ∥Po ∥

and r · ∥P − Po ∥ + ∥Po ∥ ≤ ∥P∥ for every P ∈ P (X, L(U, R)) ,

where the constant r > 0 is independent of P ∈ P (X, L(U, R)). Lemma 4.2. Suppose Y ⊆ X is a closed subspace of a normed space X . Assume P ∈ P(X, Y ) is a projection. If ∥P∥ = 1, then Y ⊥B ker P. This lemma is very useful but the proof is not difficult, so we omit it. We are ready to give the proof of the main result of this section. Theorem 4.3. Let U, Z be finite-dimensional Banach spaces. Suppose that U is strictly convex. Let X be a finite dimensional Banach space such that L(U, Z ) ⊊ X ⊆ HO(U, Z ). Then 1 < ∥P∥ for any projection P ∈ P (X, L(U, Z )). Proof. Assume, contrary to our claim, that there exists P ∈ P(X, L(U, Z )) such that 1 = ∥P∥. By Lemma 4.2 we already know that L(U, Z )⊥B ker P.

(4.1)

Moreover, ker P ̸= {0}. So there is a homogeneous mapping F ∈ HO(U, Z ) such that F ∈ X ∩ ker P and F ̸= 0. Thus there is a vector e ∈ U such that ∥e∥ = 1 and F(e) ̸= 0. Note that, since U is strictly convex, there is a ∗ in J (e) (called an exposing linear functional) such that 1 = a ∗ (e) > a ∗ (x) if x ∈ BU \ {e}. Let us define z 1 := F(e). Thus z 1 ̸= 0. Let us now define T ∈ L(U, Z ) by T u := a ∗ (u)z 1 , where u ∈ U . It is easy to check that {x ∈ BU : ∥T x∥ = ∥T ∥} = {e, −e}.

(4.2)

From (4.1) we have T ⊥B F. Due to (1.1) there exists ϕ ∈ J (T ) = { f ∈ X ∗ : ∥ f ∥ = 1, f (T ) = ∥T ∥} such that ϕ(F) = 0.

(4.3)

Now we prove that ExtJ (T ) ⊆ ExtB X ∗ . The set J (T ) is nonempty, convex, and weak*-closed, and so ExtJ (T ) ̸= ∅ by the Krein–Milman Theorem. Fix η ∈ ExtJ (T ). We show that η is an

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extreme point of the closed ball B X ∗ . Assume that η = βϑ1 + (1 − β)ϑ2 , where ϑ1 , ϑ2 ∈ B X ∗ and β ∈ (0, 1). It follows that ∥T ∥ = η(T ) = βϑ1 (T ) + (1 − β)ϑ2 (T ), from which we obtain |ϑ1 (T )| = |ϑ2 (T )| = ∥T ∥. Therefore η(T ), ϑ1 (T ) and ϑ2 (T ) are scalars in the set {−∥T ∥, ∥T ∥}, and since one is a convex combination of the others, they must all be the same scalar. Namely, ∥T ∥ = η(T ) = ϑ1 (T ) = ϑ2 (T ). Hence, ϑ1 , ϑ2 are both in J (T ), and since η ∈ ExtJ (T ), we have η = ϑ1 = ϑ2 . Thus we have proved ExtJ (T ) ⊆ ExtB X ∗ .

(4.4)

By C(BU , Z ) we denote the space of all continuous functions g from BU to Z . We now associate HO(U, Z ) with a subspace of C(BU , Z ). Namely, for each A ∈ HO(U, Z ) let Γ : HO(U, Z ) → C(BU , Z ) be defined by Γ (A) := A| BU . Then Γ is a linear isometry. Therefore, we may write L(U, Z ) ⊆ X ⊆ HO(U, Z ) ⊆ C(BU , Z ) and { } ExtB X ∗ ⊆ f | X : f ∈ ExtBC(BU ,Z )∗ . (4.5) This inclusion follows from the fact that an extreme point of ExtB X ∗ must be restriction to X of an extreme point of BC(BU ,Z )∗ (see [3, p. 257] and [11]). For t ∈ BU , let ψt : C(BU , Z ) → Z denote the evaluation function on C(BU , Z ) defined by ψt (A) := A(t). The Brosowski–Deutsch Theorem (cf. [3, Lemma 3.3]) implies that { } ExtBC(BU ,Z )∗ = y ∗ ◦ ψt ∈ BC(BU ,Z )∗ : y ∗ ∈ ExtB Z ∗ , t ∈ BU . (4.6) We conclude from (4.4), (4.5), (4.6), that there are λ1 , . . . , λm > 0, y1∗ , . . . , ym∗ ∈ ExtB Z ∗ and t1 , . . . , tm ∈ BU such that y1∗ ◦ ψt1 , . . . , ym∗ ◦ ψtm ∈ J (T ) and m ∑

λk = 1

and

ϕ=

∑m

k=1

λk · (yk∗ ◦ ψtk ).

k=1 y1∗ ◦

Since ψt1 , . . . , ym∗ ◦ ψtm ∈ J (T ), it follows that y1∗ (T t1 ) = ∥T ∥, . . ., ym∗ (T tm ) = ∥T ∥. We see at once that ∥T t1 ∥ = ∥T ∥, . . . , ∥T tm ∥ = ∥T ∥. From (4.2) we have t1 , . . . , tm ∈ {e, −e}. Directly from the definition of T , we have T (±e) = ±T (e) = ±z 1 = ±F(e) = F(±e). This clearly forces T (tk ) = F(tk ). Hence m m ∑ ∑ 0 ̸= ∥T ∥ = ϕ(T ) = λk · (yk∗ ◦ ψtk )(T ) = λk · yk∗ (T tk ) k=1

=

m ∑ k=1

λk · yk∗ (F(tk )) =

k=1 m ∑

(4.3)

λk · (yk∗ ◦ ψtk )(F) = ϕ(F) = 0.

k=1

This contradiction shows that 1 < ∥P∥.

□

It is worth saying that Theorem 4.3 cannot be generalized. Indeed, we show that the convexity assumption in Theorem 4.3 is essential. (n ) Proposition Banach ( n ) 4.4. Let X be an n + 1-dimensional ( ( n ))space such that L l1 , R ⊊ X ⊆ HO l1 , R . Then there is a projection P ∈ P X, L l1 , R such that ∥P∥ = 1. ( ) n n Proof. Note that we can write L l1n , R = l∞ and so l∞ ⊊ X . Applying the Hahn–Banach n Theorem (n times) we get immediately 1 = λ(l∞ , X ). □ We give an example to illustrate applications of Theorems 3.10 and 4.3. Namely, we prove (in some circumstances) that each minimal projection Po ∈ P (X, U ∗ ) is a strongly unique minimal projection.

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Theorem 4.5. Let U be a two-dimensional strictly convex normed space. Let X be a threedimensional normed space such that U ∗ ⊊ X ⊆ HO(U, R). Then there is exactly one minimal projection Po : X → U ∗ . Moreover, there exists r > 0 such that for every P ∈ P (X, U ∗ ): r · ∥P − Po ∥ + ∥Po ∥ ≤ ∥P∥,

(4.7)

i.e. Po is a SUM projection. Proof. Since dim X < ∞, there is a minimal projection Po : X → U ∗ . We show that Po is a SUM projection. It is obvious that U ∗ = L(U, R). Theorem 4.3 now yields 1 < ∥Po ∥ = λ(U ∗ , X ). Moreover, since dim U ∗ = 2, U ∗ has property (IE). Since U is strictly convex, ExtB(U ∗ )∗ = S(U ∗ )∗ . Thus ExtB(U ∗ )∗ is closed. By Theorem 3.10, Po is a SUM projection. Thus (4.7) is proved. □ References [1] C. Alsina, J. Sikorska, M.S. Tomás, Norm Derivatives and Characterizations of Inner Product Spaces, World Scientific, Hackensack, NJ, 2010. [2] G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J. 1 (1935) 169–172. [3] B. Brosowski, F. Deutsch, On some geometric properties of suns, J. Approx. Theory 10 (1974) 245–267. [4] S.S. Dragomir, Semi-Inner Products and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2004. [5] R.J. Fleming, J.E. Jamison, Isometries on banach spaces: function spaces, in: Monographs and Surveys in Pure and Applied Mathematics 129, 2002. [6] R.C. James, Orthogonality in normed linear linear spaces, Duke Math. J. 12 (1945) 291–301. [7] R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947) 265–292. [8] G. Lewicki, Strong unicity criterion in some space of operators, Comment. Math. Univ. Carolin. 34 (1) (1993) 81–87. [9] G. Lewicki, A. Micek, Codimension-one minimal extensions onto haar subspaces, J. Approx. Theory 164 (2012) 1461–1471. [10] G. Lewicki, M. Prophet, Codimension-one minimal projections onto haar subspaces, J. Approx. Theory 127 (2004) 198–206. [11] I. Singer, Sur l’extension des fonctionnelles linéaires, Rev. Math. Pures Appt. 1 (1956) 99–106. [12] P. Wójcik, Gateaux derivative of the norm in K(X ; Y ), Ann. Funct. Anal. 7 (4) (2016) 678–685. [13] P. Wójcik, Birkhoff orthogonality in classical M-ideals, J. Aust. Math. Soc. 103 (2017) 279–288.