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On best approximations in Banach spaces from the perspective of orthogonality

We study best approximations in Banach spaces via Birkhoff-James orthogonality of functionals. To exhibit the usefulness of Birkhoff-James orthogonality techniques in the study of best approximation problems, some algorithms and distance formulae are presented. As an application of our study, we obtain some crucial inequalities, which also strengthen the classical H\"{o}lder's inequality. The relevance of the algorithms and the inequalities are discussed through concrete examples.

Approximation theory is an extensive field of research due to its diversified applications in many branches of Science. Given any element x ∈ X and a subspace Y of X, distance between x and Y, denoted by dist(x, Y), is defined by dist(x, Y) := inf{ x − y : y ∈ Y}. An element y 0 ∈ Y is said to be a best approximation to x out of Y if dist(x, Y) = x − y 0 . The existence and the uniqueness of the best approximation cannot be guaranteed, in general. However, the existence of best approximation(s) is evidently assured for finite-dimensional subspaces. Moreover, in case of a strictly convex Banach space, the best approximation is unique, whenever it exists.
Birkhoff-James orthogonality is of essential importance in understanding the geometry of a Banach space [1,2,6,7,8]. Given any two elements x, y ∈ X, x is said to be Birkhoff-James [2,6,7] orthogonal to y, written as x ⊥ B y, if x + λy ≥ x ∀ λ ∈ R. It is not difficult to see that y 0 ∈ Y is a best approximation to x out of Y if and only if (x − y 0 ) ⊥ B Y. Given any non-zero element x ∈ X, a member f ∈ X * is called a support functional of B X (θ, x ) at x, if f = 1 and f (x) = x . The point x is called smooth if the support functional of B X (θ, x ) at x is unique. Bhatia andŠemrl completely characterized Birkhoff-James orthogonality of matrices in [1]. Based on this rudimentary result, the authors provided some distance formulae in the same article. One may consult [5,10] for a study of best approximations and orthogonality of matrices. We refer the readers to [11,14,17,19] for some current works involving the geometry of operator spaces and orthogonality of operators in Banach space setting. Some recent developments on best approximations to compact operators can be found in [16], where the central themes are semi-inner-products [4,9,15] and operator orthogonality.
The current article presents a comprehensive approach to address the problem of finding best approximation(s) to a given point x out of a subspace Y, in its full generality. After recalling some basic facts in Section 2, we build up the theoretical background of our work in Section 3 and Section 4. The results presented in Section 4 should be viewed as generalizations of the results obtained in [16] and will be mentioned accordingly. The integral theme of our development is Birkhoff-James orthogonality of functionals. Application of Birkhoff-James orthogonality not only reduces the computational difficulties to resolve the above mentioned problem but also strengthens the classical duality principle [16,Section 4]. An extra advantage of employing the concept of Birkhoff-James orthogonality (over that of the classical duality principle) is that it provides an easy way out to compute the all possible best approximation(s) to x out of Y. Indeed, we devote Section 5 to show the applicability of the results, developed in the preceding sections, in context of the said problem. We obtain concrete solutions to some problems regarding best approximations and provide certain distance formulae under specific assumptions, which also produce some interesting inequalities, including a finite-dimensional strengthening of the classical Hölder's inequality.

Preliminaries
In this section, we mention some known facts that will be used extensively in the next two sections. We begin with a simple proposition which has important applications in the study of topological vector spaces.
Suppose that τ is a topology on a vector space X such that every one point set in X is closed and the vector space operations on X are continuous with respect to the topology τ . Then the vector space X equipped with the topology τ is called a topological vector space. The topological vector space X is called locally convex if there exists a local base at θ, whose members are convex. Every topological vector space enjoys an important separation property: Proposition 2.2. [12, Theorem 1.10] Let X be a topological vector space. Let K and C be subsets of X such that K is compact and C is closed with K ∩ C = ∅. Then there exists a neighborhood V of θ such that Given a Banach space X, X * equipped with the weak * topology is a locally convex topological vector space. Moreover, every linear functional on X * that is weak * continuous is of the form ψ(x) for some x ∈ X, where ψ denotes the canonical embedding of X into X * * . We refer the readers to the standard text [12] for more information in this regard. Weak * topology on X * has a crucial compactness property known as the Banach-Alaoglu Theorem: Theorem 2.3. (Banach-Alaoglu) Let X be a normed linear space. Then the closed unit ball B X * of X * is compact with respect to the weak * topology on X * .
We next present a classical result which is a variant of the geometric Hahn-Banach Theorem and is popularly known as the Mazur Theorem. In the following theorem, we do not require the topological vector space to be locally convex.
Then there exists a closed hyperplane H of X such that Let X be a Banach space and let W be any non-trivial subspace of X. Let f be any member of X * . A member f 0 of X * is said to be a Hahn-Banach extension of The next two propositions are about some basic facts regarding real ℓ p spaces, where p ∈ (1, ∞). Given any 1 ≤ p ≤ ∞, q is said to be the conjugate to p if q = 1 (q = ∞), whenever p = ∞ (p = 1), and, q = p p−1 , whenever 1 < p < ∞. Proposition 2.5. Let p ∈ (1, ∞). Then the dual of ℓ p is isometrically isomorphic to ℓ q , and for any c = (c 1 , c 2 , . . . ) ∈ ℓ q , the corresponding member f c ∈ ℓ * p is given by: Proposition 2.6. Let p ∈ (1, ∞) and let a = (a 1 , a 2 , . . . ) ∈ ℓ p be non-zero. Let c = (c 1 , c 2 , . . . ) ∈ ℓ q be such that c corresponds to the support functional of a in ℓ * p . Then for each k ∈ N, c k is given by:

Orthogonality of functionals
Our aim in this section is to obtain a characterization of Birkhoff-James orthogonality of weak * continuous functionals, which is of paramount importance in the context of our current work on best approximations. We would like to mention that the said characterization can also be obtained by modifying Theorem 2.1 of [13]. However, we present a complete proof of the same, for the convenience of the readers. We need the following proposition, which completely describes the norm attainment sets of weak * continuous functionals, to serve our purpose. Proposition 3.1. Let X be a Banach space and let f ∈ X * * be weak * continuous. Then M f = ±D, where D is a non-empty compact path connected subset of B X * with respect to the weak * topology on X * .
Proof. Denote the canonical embedding of X into X * * by ψ. Since f is weak * continuous, there exists x 0 ∈ X such that ψ(x 0 ) = f . By the Hahn-Banach Theorem, there exists a support functional of B X (θ, x 0 ), say Therefore, M f = ∅. Let D be a subset of B X * , defined by: Evidently, M f = ±D and D is non-empty since M f is non-empty. Also, it is easy to see that D is a closed subset of B X * . It now follows from the Banach-Alaoglu Theorem that D is a compact subset of B X * , with respect to the weak * topology on X * . Note that for any 1]. Since X * equipped with the weak * topology is a topological vector space, therefore, the mapping t → tx * 1 + (1 − t)x * 2 is continuous. Consequently, D is a path connected subset of B X * . This completes the proof.
Based on Proposition 3.1, we now obtain a characterization of orthogonality of weak * continuous functionals. Note that the following characterization also includes [16,Theorem 3.2].
Theorem 3.2. Let X be a Banach space and let f, g ∈ X * * be weak * continuous.
Proof. We only prove the necessary part as the proof of the sufficient part is trivial. Suppose on the contrary that M f ∩ N (g) = ∅. By Proposition 3.1, M f = ±D, where D is a non-empty compact connected subset of B X * with respect to the weak * topology on X * . It now follows from the connectedness of D that either Obviously, p is continuous. Also, for each y ∈ M f there exist a weak * open set U y containing y and δ y ∈ (0, 1) such that On the other hand, for each z ∈ B X * \M f there exist a weak * open set V z containing z and δ z ∈ (0, 1) such that Therefore, the collection {U y : y ∈ M f } ∪ {V z : z ∈ B X * \ M f } forms a weak * open cover for B X * . Due to the compactness of B X * (with respect to the weak * topology on X * ), there exist natural numbers k 1 , k 2 such that This is a contradiction to the fact that f ⊥ B g. This completes the proof.
As an application of Theorem 3.2 we have the following corollary: Proof. Denote the canonical embedding of X into X * * by ψ. Let ψ(x) = f and ψ(y) = g. Then x ⊥ B y if and only if f ⊥ B g. Since f, g ∈ X * * are weak * continuous, it follows from Theorem 3.
This completes the proof.

Birkhoff-James orthogonality and best approximations
Let X be a reflexive, strictly convex Banach space. One of the fundamental ideas in [16] was to identify the Banach space X with its double dual X * * and then treat the best approximation problem in X * * by employing Birkhoff-James orthogonality techniques. Unfortunately, the above idea does not work if X is not reflexive. This lacuna can be overcome by identifying X to the space of all weak * continuous functionals on X * . The following theorem provides a necessary and sufficient condition regarding the best approximation problem in the space of all weak * continuous functionals. In that sense, the result is a generalization of Theorem 3.4 and Theorem 3.5 of [16]. Also, note that we do not require the strict convexity of X.
Proof. We first prove the necessary part. Suppose on the contrary that there exists a finite-dimensional subspace Z of Y containing g 0 such that it follows from Proposition 3.1 that M f −g0 = ±D, where D is a compact convex subset of B X * with respect to the weak * topology on X * . Since X * equipped with the weak * topology is a locally convex topological vector space and Since D is convex, so is D + V. In particular, D + V is an open convex subset of X * , disjoint from g∈Z N (g). Therefore, it follows from Theorem 2.4 that there exists a closed hyperplane H of X * such that Let h : X * → R be a linear functional such that N (h) = H. It now follows from Proposition 2.1 that h ∈ span {h j : 1 ≤ j ≤ k} ⊆ Y. In particular, h is weak * continuous. Also, note that However, this is a contradiction to the fact that g 0 is a best approximation to f out of Y.
To prove the sufficient part of the theorem, let g 1 ∈ Y be arbitrary and let Z = span{g 0 , g 1 }. Then it follows from the hypothesis of the theorem that Whenever Y is finite-dimensional, the above theorem takes a simpler form. We record this as a corollary. The proof of the corollary follows directly from Theorem 4.1, and therefore, it is omitted.
As an immediate application of Theorem 4.1, we now obtain a distance formula in the space of all weak * continuous functionals on X * . The following distance formula can be regarded as a strengthened version of Theorem 3.6 of [16].
Theorem 4.2. Let X be a Banach space and let f ∈ X * * be weak * continuous. Let Y be a subspace of X * * such that each member of Y is weak * continuous and f / ∈ Y. Suppose that g 0 ∈ Y is a best approximation to f out of Y. Then for any finite-dimensional subspace Z of Y containing g 0 , Proof. It follows from Theorem 4.1 that Then it is easy to see that (1)).
We now claim that which is a contradiction. This completes the proof.
Assuming Y to be finite-dimensional in the above theorem, we have the following distance formula: Corollary 4.2.1. Let X be a Banach space and let f, g 1 , g 2 , . . . , g m ∈ X * * be weak * continuous for some m ∈ N. Let g 1 , g 2 , . . . , g m be linearly independent and let Then as in Theorem 4.2, we get the desired formula.
As mentioned in the introduction, Birkhoff-James orthogonality techniques provide some genuine insights in determining best approximation(s) to a given point out of a subspace. Indeed, the following result completely characterizes best approximation(s) to a given point out of a finite-dimensional subspace in the setting of weak * continuous functionals. Theorem 4.3. Let X be a Banach space and let f, f 0 , g 1 , g 2 , . . . , g m ∈ X * * be weak * continuous for some m ∈ N. Let g 1 , g 2 , . . . , g m be linearly independent and Proof. We first prove the necessary part. Suppose on the contrary that which is absurd. Therefore, M f0 ∩ W = ∅, as expected. Then Corollary 4.1.1 ensures that f −f 0 is not a best approximation to f out of Y, which is a contradiction.
We now prove the sufficient part.
Thus, y 1 ∈ M f0 and M f W ⊆ M f0 . Consequently, M f0 ∩W = ∅. Therefore, applying Corollary 4.1.1, we obtain that f − f 0 is a best approximation to f out of Y. This completes the proof.
We would like to remark here that the above theorem can also be stated in terms of norm attainment sets of weak * continuous functionals.
Remark 4.1. Let X be a Banach space and let f, g 1 , g 2 , . . . , g m ∈ X * * be weak * continuous for some m ∈ N. Let g 1 , g 2 , . . . , g m be linearly independent and let f / ∈ Y, where Y = span{g 1 , g 2 , . . . , g m }. Let us consider the set B := {h ∈ X * * : f − h is a best approximation to f out of Y}.
Then B is precisely the collection of those weak * continuous functionals which are and whose norm attainment sets contain the norm attainment set of the restriction of f to In other words, for any h ∈ B, It is obvious that given any Banach space X, the best approximation problems in X can be treated as the best approximation problems in the space of all weak * continuous functionals on X * . Therefore, the foregoing results can also be rephrased in terms of best approximation problems in X. As an evidence of this, we present the following theorem which is essentially a variant of Theorem 4.1. However, we refrain ourselves from doing analogous treatment to the remaining results, to avoid monotony.
Theorem 4.4. Let X be a Banach space and let x 0 ∈ X. Let Y be a subspace of X such that x 0 / ∈ Y and y 0 ∈ Y. Then y 0 is a best approximation to x 0 out of Y if and only if for every finite-dimensional subspace Z of Y containing y 0 , z∈Z N (ψ(z)) M ψ(x0−y0) = ∅, where ψ : X → X * * denotes the canonical embedding.
We have seen in Theorem 3.2 that there is a deep connection between Birkhoff-James orthogonality and the norm attainment set of a given functional. Since every weak * continuous functional is norm attaining, there is a scope to employ Birkhoff-James orthogonality techniques in all of the preceding results. However, Birkhoff-James orthogonality is not so straightforward for the functionals that do not attain their norms. Consequently, the same techniques cannot be applied for functionals not attaining their respective norms. We end this section with a result which addresses this issue. Also, note that the result is valid in any Banach space, real or complex.
Theorem 4.5. Let X be a Banach space and let f ∈ X * . Let Y be a subspace of X * such that f / ∈ Y and g 0 ∈ Y. Then g 0 is a best approximation to f out of Y if and only if for every finite-dimensional subspace Z of Y containing g 0 , the following holds true: Proof. We first prove the necessary part. Suppose on the contrary that there exists a finite-dimensional subspace Z of Y containing g 0 such that Let {h j : 1 ≤ j ≤ k} be a basis of Z. Then it is straightforward to check that By the Hahn-Banach Theorem, f 0 possesses a linear extension f 0 : X → R such that .

Now, consider the linear functional
where the second last equality follows from the fact that f (x) = (f − g 0 )(x) for all x ∈ g∈Z N (g). However, this is a contradiction to the fact that g 0 is a best approximation to f out of Y.
We now prove the sufficient part. Let g 1 ∈ Y be arbitrary and let Z = span{g 0 , g 1 }. Then it follows from the hypothesis of the theorem that Thus, g 0 is a best approximation to f out of Y and this completes the proof.

Some applications and distance formulae
In this section we exhibit some interesting applications and examples to the theories developed in the preceding sections. Let us begin with an algorithm that generalizes the Problem given in [16].
Problem 5.1. Let X be a Banach space and let m ∈ N. Let x 0 , y 1 , y 2 , . . . , y m ∈ X be such that y 1 , y 2 , . . . , y m are linearly independent and x 0 / ∈ Y, where Y = span{y 1 , y 2 , . . . , y m }. Then find best approximation(s) to x 0 out of Y and compute dist(x 0 , Y). Step 1: We embed X into its double dual X * * via the canonical isometric isomorphism ψ. Let ψ(x 0 ) = f and ψ(y i ) = g i for 1 ≤ i ≤ m. Let Z = span{g 1 , g 2 , . . . , g m }. Evidently, the above problem is equivalent to finding the best approximation(s) to f out of Z and computing dist(f, Z). Since f, g 1 , g 2 , . . . , g m are weak * continuous, the criteria of Corollary 4.1.1, Corollary 4.2.1 and Theorem 4.3 are satisfied.
Step 2: Let W = m i=1 N (g i ). We now consider the following two cases: Case I: W is one-dimensional. Consider any non-zero z ∈ W. Then it follows from Corollary 4.2.1 that Case II: W is not one-dimensional. Consider W ∩ kerf and find some non-zero u ∈ W such that u ⊥ B (W ∩ kerf ). Note that the existence of such an u is always guaranteed, since M f W = ∅. Then we have that In other words, Step 3: Let us consider the following collection: Λ := f − f 0 ∈ X * * : f 0 is a Hahn-Banach extension of f W .
It is not difficult to see that Λ = ∅ and Λ ⊆ Z. It follows from Theorem 4.3 that Λ is precisely the collection of best approximation(s) to f out of Z.
Thus, we completely obtain the solution of the above problem, as ψ −1 (Λ) is precisely the collection of best approximation(s) to x 0 out of Y.
The algorithm presented in Problem 5.1 is particularly advantageous for ℓ n 1 and ℓ n ∞ spaces, for n ∈ N. This is because the dual of ℓ n 1 (ℓ n ∞ ) is ℓ n ∞ (ℓ n 1 ) and if any member x * of ℓ n * 1 (ℓ n * ∞ ) corresponds to a member (a 1 , a 2 , . . . , a n ) of ℓ n ∞ (ℓ n 1 ), then the action of x * on any member (b 1 , b 2 , . . . , b n ) of ℓ n 1 (ℓ n ∞ ) is given by the formula: We elaborate this in more detail in the following problem: Problem 5.2. Let X = ℓ n 1 or ℓ n ∞ for some n ∈ N. Let x 0 , y 1 , y 2 , . . . , y m ∈ X, where 1 ≤ m < n be such that y 1 , y 2 , . . . , y m are linearly independent and x 0 / ∈ Y, where Y = span{y 1 , y 2 , . . . , y m }. Compute dist(x 0 , Y).
Step 2: Note that Therefore, applying Corollary 4.2.1, we obtain In other words, the problem of computing dist(x 0 , Y) reduces to the problem of finding the absolute maximum of f on the unit sphere of the solution space of the system of linear equations: Note that the non-triviality of the solution space is guaranteed by the existence of best approximation(s).
We now present an example to illustrate the utility of the above problem: 0, 2, 0). Let x 0 = (1, 1, 1, 1). Then it is trivial to see that x 0 / ∈ Y. Our aim is to calculate dist(x 0 , Y) and to find a best approximation to x 0 out of Y.
We now follow the procedure, as described in Case II of Problem 5.1. Therefore, we find Next, we find a best approximation to f out of Z. Define f 0 : Let f 0 : ℓ 4 ∞ → R be defined by Then we have that . Therefore, it follows from Theorem 4.3 that f − f 0 is a best approximation to f out of Z. Thus, 1, 1, 0) is a best approximation to x 0 out of Y.
In light of Problem 5.2, we can say that best approximation problems in context of the ℓ n ∞ and ℓ n 1 spaces reduce to the problem of maximizing a functional to the unit sphere of the solution space of a system of homogeneous linear equations, and the problem becomes trivial if m = n − 1. Indeed, in that case all we need to do is to solve a system of homogeneous linear equations. On the other hand, we have the following explicit distance formulae whenever n = 2: Proof. Let Ω : ℓ 2 1 −→ ℓ 2 * ∞ be the canonical isometric isomorphism. Let Ω(x) = f and let Ω(y) = g. Clearly, N (g) = (u 1 , u 2 ) ∈ R 2 : cu 1 + du 2 = 0 .
Applying Corollary 4.2.1, we obtain that Similarly, if c = 0, one can show that This completes the proof.
Applying Corollary 4.2.1, we obtain that Similarly, if c = 0, one can show that This completes the proof.
In view of Proposition 5.3, Proposition 5.4 and Theorem 3.8 of [16], the proof of the following result is obvious.
where q is conjugate to p.
The next theorem presents a sufficient condition for the uniqueness of the best approximation in finite-dimensional real polyhedral Banach space. Recall that a finite-dimensional real Banach space is called polyhedral if Ext(B X ) is finite. For more information on the local structure of finite-dimensional real polyhedral Banach spaces, we refer the readers to [18].
Theorem 5.6. Let X be an n-dimensional real polyhedral Banach space and let x, y 1 , y 2 , . . . , y m ∈ X; 1 ≤ m < n. Let y 1 , y 2 , . . . , y m be linearly independent and x / ∈ Y, where Y = span{y 1 , y 2 , . . . , y m }. Let ψ : X → X * * denote the canonical isometric isomorphism. Suppose that m i=1 N (ψ(y i )) S X * contains only smooth point(s) of B X * . Then the best approximation to x out of Y is unique.
Proof. Let ψ(x) = f and let ψ(y i ) = g i for each 1 ≤ i ≤ m. Clearly, f / ∈ Z, where Z = span{g 1 , g 2 , . . . , g m }. It is enough to show that the best approximation to f out of Z is unique. Suppose that m i=1 α i g i and m i=1 β i g i are best approximations to f out of Z for real numbers α 1 , α 2 , . . . , α m ; β 1 , β 2 , . . . , β m . Note that non-smooth point of B X * . Let g 0 be a best approximation to f out of Z. Then it follows from Corollary 4.1.1 that N (g i ) S X * contains only smooth points of the facets F and −F , it follows that f − g 0 = λh, where h ∈ S X * * is the unique support functional corresponding to the facet F [18] and λ ∈ R is non-zero. However, this proves that where λ 1 , λ 2 are non-zero real numbers with |λ 1 | = |λ 2 |. Suppose that λ 1 = −λ 2 = λ 0 . Then it follows that which is a contradiction, since f / ∈ Z. Therefore, we must have α i g i and the best approximation is unique. This completes the proof of the theorem.
The converse of the above theorem need not be true. The following example illustrates such a situation: Example 5.6.1. Let X = ℓ 3 1 and let x = (0, 1 2 , 1 2 ), y = (0, 0, 1). We identify X * * to the dual of ℓ 3 ∞ and let ψ : X → X * * denote the canonical isometric isomorphism. Let ψ(x) = f and ψ(y) = g. Then it is not difficult to see that Clearly, which also contains non-smooth points of B X * . Our aim is to show that the best approximation to x out of span{y} is unique.
Let f − λ 0 g be the best approximation to f out of span{g}. Then it follows from Corollary 4.2.1 that Next, we define f 0 : Now, it is not difficult to see that Consequently, f 0 is a Hahn-Banach extension of f N (g) . Therefore, it follows from Theorem 4.3 that f − f 0 = 1 2 g is a best approximation to f out of span{g}.
For any λ ∈ R, consider the linear functional (f − λg) : ℓ 3 ∞ → R, given by: Therefore, the only solution of λ 0 for which f − λ 0 g = 1 2 is λ 0 = 1 2 and the best approximation to f out of span{g} is unique. Consequently, the best approximation to x out of span{y} is unique.
As an application of the ideas developed in Theorem 4.1 and Theorem 4.2.1, it is also possible to explore the following invariant distance problem: Problem 5.7. Let (X 1 , · 1 ) and (X 2 , · 2 ) be two Banach spaces with X 1 ⊆ X 2 . Let Y be a finite-dimensional vector subspace of X 1 ∩ X 2 and let x 0 ∈ (X 1 ∩ X 2 ) \ Y. Then find a necessary and sufficient condition on x 0 and Y such that In the following theorem, we provide a complete solution to the above problem when X 1 = ℓ p1 and X 2 = ℓ p2 , where 1 < p 1 , p 2 < ∞ and p 1 = p 2 .
It can be seen from [16, Remark 3.10] that best approximation problems give rise to a family of inequalities in context of ℓ p spaces (1 < p < ∞). Our next goal is to find the said family of inequalities in a more general setting. The following result is the first step towards achieving the said goal.
Theorem 5.9. Let n ∈ N and let X i = ℓ mi pi , where m i are natural numbers and Let n i=1 m i = m and let T be an m by m non-zero matrix. Then for any x 0 ∈ X the following holds true: Proof. We begin the proof with the observation that the existence of the minimum of the set { x 0 − y X : y ∈ R(T )} is guaranteed by the existence of best approximation(s) to x 0 out of R(T ). Let q, q 1 , q 2 , . . . , q n be the conjugates to p, p 1 , p 2 , . . . , p n , respectively. Clearly, for each 1 ≤ i ≤ n, Also, for any member x * = (x * 1 , x * 2 , . . . , x * n ) ∈ X * , Clearly, X and R m are isomorphic as vector spaces. Let {e j } m j=1 denote the standard ordered basis of X. Observe that R(T ) = span{y 1 , y 2 , . . . , y m }, where y j = T (e j ) for all 1 ≤ j ≤ m. Therefore, . Let x 0 = (u 1 , u 2 , . . . u n ), where u i ∈ X i for all 1 ≤ i ≤ n. Also, let Let Ω : Then it is not difficult to see that for each 1 ≤ j ≤ m, Let Z = span{g 1 , g 2 , . . . , g m }. Note that In other words, m j=1 N (g j ) = z ∈ R m : [y t 1 y t 2 . . . y t m ] t z = θ = {z ∈ R m : T * z = θ} = N (T * ).
Moreover, the above inequality is optimal.
Proof. Let H denote the Hilbert space R m equipped with the usual dot product. Let Y = span{(a 1 , a 2 , . . . , a m )}. Therefore, we have that W = Y ⊥ . Let for each 1 ≤ i ≤ n X i = ℓ mi pi and X = n i=1 p X i .
Then X and R m are isomorphic as vector spaces. Let {e j } m j=1 denote the standard ordered basis of X. Define T : H → H in such a way that R(T ) = span{T (e j ) : 1 ≤ j ≤ m} = Y.
Then it is easy to see that N (T * ) = Y ⊥ . It now follows from Theorem 5.9 that min{ x 0 − y X : y ∈ R(T )} = max 1 z X * | x 0 , z | : z ∈ N (T * ) \ {θ} , Now, inequality (5) is obtained by removing minimum and maximum from both sides, and expressing x 0 − y X and z X * , in their explicit forms. Moreover, the inequality is optimal. This completes the proof.
The inequality (5) obtained in Theorem 5.10 exhibits a stronger version of Hölder's inequality in finite-dimensional case. We explain this in more detail in the following remark: