Coproximinality of linear subspaces in generalized Minkowski spaces
aa r X i v : . [ m a t h . M G ] J a n COPROXIMINALITY OF LINEAR SUBSPACES INGENERALIZED MINKOWSKI SPACES
THOMAS JAHN AND CHRISTIAN RICHTER
Abstract.
We show that, for vector spaces in which distance measurementis performed using a gauge, the existence of best coapproximations in -codimensional closed linear subspaces implies in dimensions ≥ that the gaugeis a norm, and in dimensions ≥ that the gauge is even a Hilbert space norm.We also show that coproximinality of all closed subspaces of a fixed dimensionimplies coproximinality of all subspaces of all lower finite dimensions. Introduction
A basic approximation task in Banach spaces X requires to “replace” an element y ∈ X by an element x of a subset K ⊂ X subject to some notion of proximity.This can be implemented by the notion of best approximation, a vivid researchtopic whose Euclidean subcase is a common topic of undergraduate calculus oroptimization courses. In the theory of best approximation by elements of linearsubspaces (or, more generally, convex subsets) K ⊂ X , see Singer’s monograph[20], this notion gives rise to a set-valued operator P K : X ⇒ X , and one is in-terested in characterizations of proximinal sets and of Chebyshev sets , i.e., sets K ⊂ X for which P K ( x ) is non-empty or a singleton for all x ∈ X , respectively.Franchetti and Furi [7] introduced a companion notion called best coapproximation in the setting of normed spaces, yielding another set-valued operator Q K : X ⇒ X .Notions of coproximinal sets and coChebyshev sets can be defined and investigatedanalogously, see [1]. Berens and Westphal [5] show that a coChebyshev subset K of a Hilbert space H is necessarily a translate of a closed linear subspace. Pap-ini and Singer [18] link best coapproximations to Birkhoff orthogonality, providecharacterizations in terms of linear functionals, give examples in normed spaces ofreal-valued continuous functions defined on an interval, and discuss properties ofbest coapproximations viewed as a set-valued operator. (In the context of these re-sults, also [9, Theorem 3.2] has to be mentioned, where it is shown that in Banachspaces of dimension ≥ , symmetry of the Birkhoff orthogonality relation alreadyimplies Hilbertianity.) Franchetti and Furi [7, Lemma 1(d) and Theorem 1] showthat each closed -codimensional linear subspace K of a Banach space X of dimen-sion ≥ is coproximinal if and only if X is a Hilbert space, and that straight linesare always coproximinal in normed spaces.The contributions of the present paper are motivated by these very results of [7], andthey take place in the much broader setting of generalized Minkowski spaces . Thoseare real vector spaces X equipped with a function γ : X → R (called a gauge ) which Mathematics Subject Classification.
Key words and phrases. best coapproximation, bisector, coproximinal, gauge, Hilbert space,Minkowski space, norm. takes only non-negative values, vanishes only at ∈ X , and meets the conditions γ ( λx ) = λγ ( x ) and γ ( x + y ) ≤ γ ( x )+ γ ( y ) for all λ > and x, y ∈ X . This way, everynorm on X is a gauge, but not vice versa. (Norms are precisely the gauges γ forwhich γ ( x ) = γ ( − x ) .) Nonetheless, basic concepts from classical functional analysiscan be set up in generalized Minkowski spaces in an analogous way, see Cobzas , ’smonograph [6]. The number γ ( x − y ) is still interpreted as a distance from y to x ,although in the gauge distance it need not coincide with the distance from x to y anymore. This enables the translation of distance-based concepts of approximationtheory to generalized Minkowski spaces. Apart from functional analysis [2] andapproximation theory [3], the study of generalized Minkowski spaces, which datesback to Minkowski [17], has also attracted researchers in location science [19] andconvex geometry [16].Combining the setting of generalized Minkowski spaces and the notion of best coap-proximation, we show in Theorem 3.2 that coproximinality of straight lines charac-terizes normed spaces among generalized Minkowski spaces in dimensions ≥ , andextend the characterization of Hilbert spaces of dimension ≥ via coproximinalityof -codimensional linear subspaces to all generalized Minkowski spaces in Theo-rem 5.1. Note that we allow for infinite-dimensional vector spaces X in this paper.This was not the case in the authors’ past papers on generalized Minkowski spacessuch as [12]. 2. Background and notation
The following definition is taken from [11, Definition 3.10].
Definition 2.1.
Let ( X, γ ) be a generalized Minkowski space and K ⊂ X . A point x ∈ K is called a best coapproximation of y ∈ X in K if γ ( x − z ) ≤ γ ( y − z ) for all z ∈ K . The set of all best coapproximations of y in K shall be denoted by Q K ( y ) .The authors of [1] introduce the notion of coproximinal sets , meaning sets K forwhich Q K ( x ) is non-empty for all x ∈ X . We shall use this term also in the settingof generalized Minkowski spaces.In [15], the bisector bsc γ ( x, y ) = { z ∈ X | γ ( z − x ) = γ ( z − y ) } has been intro-duced to the setting of generalized Minkowski spaces and investigated in the caseof dim( X ) ∈ { , } . Clearly, knowing the shape of the bisector bsc γ ∨ ( x, y ) takenwith respect to the gauge γ ∨ : X → R defined by γ ∨ ( x ) := γ ( − x ) is of interest forthe study of best coapproximation.The study of gauges is determined by the geometry of their balls B γ ( y, λ ) := { x ∈ X | γ ( x − y ) ≤ λ } , U γ ( y, λ ) := { x ∈ X | γ ( x − y ) < λ } , and spheres S γ ( y, λ ) := { x ∈ X | γ ( x − y ) = λ } . Balls are convex sets, i.e., sets which contain the whole line segment [ x, y ] := { x + λ ( y − x ) | ≤ λ ≤ } with each two points x and y they contain. We shall usethe notations h x, y i for the straight line passing through points x, y ∈ X , [ x, y i := { x + λ ( y − x ) | λ ≥ } for the ray starting at x and passing through y , cone ( A ) := { λx | x ∈ A, λ > } for the conical hull of a set A ⊂ X , and lin ( A ) for its linearhull . For A , A ⊂ X and x ∈ X , we write A ± A := { a ± b | a ∈ A , b ∈ A } and x + A = A + x := { x } + A . A linear subspace H of X is said to be -codimensional if there exists a vector v ∈ X \ H such that X = { x + αv | x ∈ H, α ∈ R } . OPROXIMINALITY OF LINEAR SUBSPACES IN GENERALIZED MINKOWSKI SPACES 3
Using the notion of balls, one may write for the set of best coapproximations Q K ( y ) = K ∩ \ z ∈ K B γ ( z, γ ( y − z )) = K ∩ \ z ∈ K,λ> y ∈ B γ ( z,λ ) B γ ( z, λ ) . Characterizing normed spaces in terms of coproximinality
Coproximinality of all straight lines characterizes -dimensional normed spacesamong the -dimensional generalized Minkowski spaces. Lemma 3.1.
Let ( X, γ ) be a generalized Minkowski space with dim( X ) = 2 . Thefollowing statements are equivalent.(i) The gauge γ is a norm.(ii) Every straight line K ⊂ X is coproximinal.Proof. For the implication (i) ⇒ (ii), see [7, Lemma 1(d)]. For the converse impli-cation, assume that γ is not a norm. By [21, 4.1], there exists a chord [ x , x ] of B γ (0 , such that ∈ [ x , x ] and [ x , x ] is no affine diameter of B γ (0 , , i.e., B γ (0 , has no pair of parallel supporting lines passing through x and x , respec-tively. Thus there exists another chord [ y , y ] of B γ (0 , and a number λ > such that y − y = λ ( x − x ) .Let K = h x , x i . Then , y − y ∈ K and γ ( y −
0) = 1 = γ ( y − ( y − y )) . Itfollows that Q K ( y ) = K ∩ \ z ∈ K B γ ( z, γ ( y − z )) ⊂ K ∩ ( B γ (0 , ∩ B γ ( y − y , K ∩ B γ (0 , ∩ ( K ∩ B γ ( y − y , x , x ] ∩ ([ x , x ] + y − y )= ∅ Therefore, the straight line K is not coproximinal. (cid:3) Let us give two alternative proofs of the implication (i) ⇒ (ii) in Lemma 3.1. First proof.
Let y ∈ X . Let u and u be the points of intersection of the straightline y + ( K − K ) and S γ ( y, . The convex body B γ ( y, admits a pair of parallelsupporting lines L at u and L at u . The straight lines L and L intersect K in points v and v , respectively. Take x = ( v + v ) .Case 1: L ∩ S γ ( y,
1) = { u } . Then [15, Lemma 2.1.1.1] implies that the bisector bsc γ ( x, y ) is contained in the interior of the convex hull of the union of K and y + ( K − K ) or, in other words, γ ( y − z ) > γ ( x − z ) for all z ∈ K , see Figure 1 foran illustration.Case 2: L ∩ S γ ( y,
1) = [ w , w ] . Assume that there exists λ > such that w − w = λ ( x − y ) . (Else, change the roles of w and w .) Let s be the intersection pointof [ y, w i and [ x, w + x − y i , and s be the intersection point of [ y, y − w i and [ x, x − ( w + x − y ) i . By [13, Theorem 2.1 and Proposition 2.4], the bisector bsc γ ( x, y ) can be written as a union A ∪ A ∪ A with A = (cone ([ w , w ] − y )+ y ) ∩ (cone ([ w , w ] − y ) + x ) , A = ( − cone ([ w , w ] − y ) + y ) ∩ ( − cone ([ w , w ] − y ) + x ) , and A being contained in the convex hull of { x, y, s , s } , see Figure 2 for anillustration. But then, we also have γ ( y − z ) ≥ γ ( x − z ) for all z ∈ K .In either case, x ∈ Q K ( y ) . (cid:3) THOMAS JAHN AND CHRISTIAN RICHTER L L y xu u v v K Figure 1.
Notation in Case 1 of an alternative proof of the im-plication (i) ⇒ (ii) in Lemma 3.1. L L y xw w s s u u v v A A K Figure 2.
Notation in Case 2 of an alternative proof of the im-plication (i) ⇒ (ii) in Lemma 3.1. Second proof.
Let K ⊂ X be a straight line and let y ∈ X . Coproximinality istranslation invariant, i.e., if a set K ⊂ X is coproximinal, then so is K + x forevery x ∈ X . Thus there is no loss of generality in assuming that K is a linearsubspace of X . Then K ∩ B γ (0 ,
1) = [ − x , x ] , and there is a linear functional f : X → R , f = 0 , such that max { f ( x ) | x ∈ B γ (0 , } = f ( x ) =: α > and min { f ( x ) | x ∈ B γ (0 , } = f ( − x ) = − α < .For z ∈ K and λ ≥ , we then have B γ ( z, λ ) ∩ K = f − ([ f ( z ) − λα, f ( z ) + λα ]) ∩ K ,and y ∈ B γ ( z, λ ) implies f ( y ) ∈ [ f ( z ) − λα, f ( z ) + λα ] . Hence, Q K ( y ) = K ∩ \ z ∈ K,λ> y ∈ B γ ( z,λ ) B γ ( z, λ )= \ z ∈ K,λ> y ∈ B γ ( z,λ ) (cid:0) f − ([ f ( z ) − λα, f ( z ) + λα ]) ∩ K (cid:1) ⊃ \ z ∈ K,λ> y ∈ B γ ( z,λ ) (cid:0) f − ( f ( y )) ∩ K (cid:1) = f − ( f ( y )) ∩ K = ∅ . OPROXIMINALITY OF LINEAR SUBSPACES IN GENERALIZED MINKOWSKI SPACES 5
For the latter inequality, note that the set f − ( f ( y )) is a straight line parallelto supporting lines of B γ (0 , at x and − x , so it cannot be disjoint from (i.e.,parallel to) h− x , x i = K . (cid:3) The characterization established in Lemma 3.1 translates directly to higher dimen-sions.
Theorem 3.2.
Let ( X, γ ) be a generalized Minkowski space with dim( X ) ≥ . Thefollowing statements are equivalent.(i) The gauge γ is a norm.(ii) Every straight line K ⊂ X is coproximinal.Proof. We reduce the problem to the -dimensional case and apply Lemma 3.1.First, by translational invariance of coproximinality, it is sufficient to consider -dimensional linear subspaces instead of arbitrary straight lines. Second, for thecoproximinality of a set K ⊂ X , the points of K are not crucial as we always have Q K ( y ) = { y } when y ∈ K . Third, for a set K ⊂ X and a point y ∈ X , the set Q K ( y ) stays the same when viewed in the generalized Minkowski space ( Y, γ | Y ) instead of ( X, γ ) , with Y := lin ( K ∪ { y } ) and γ | Y being the restriction of γ to Y .Taking these observations into account, the coproximinality of every straight line K ⊂ X is equivalent to the coproximinality of every -dimensional linear subspace K of X with respect to every -dimensional linear subspace ( Y, γ | Y ) of ( X, γ ) with K ⊆ Y . (cid:3) Some remarks on topology
In order to give an account on the topological arguments that come with the fol-lowing characterization of Hilbert spaces, we loosely follow the exposition of [6].For a generalized Minkowski space ( X, γ ) , we call a set V ⊂ X a neighborhood of x ∈ X , if there exists a number λ > such that U γ ( x, λ ) ⊂ V . This way, the familyof neighborhoods of a fixed point forms a filter of subsets of X . Clearly, anotherbasis of that filter is given by B γ ( x, λ ) , λ > . Those subsets of X which areneighborhoods of all of their points form the family of all open sets of a topology τ γ on X . A set F ⊂ X is closed in that topology if X \ F is open or, equivalently,if F is closed under τ γ -limits. Here the expression lim n →∞ x n = x is defined by lim n →∞ γ ( x n − x ) = 0 .Note that lim n →∞ γ ( x n − x ) = 0 does not imply lim n →∞ γ ( x − x n ) = 0 (seeExample 4.3 below). That is, the gauges γ and γ ∨ may give rise to different conceptsof convergence. In fact, two gauges generate the same topology or, equivalently,the same concepts of convergence if and only if they are equivalent in the followingsense. Proposition 4.1.
Let ( X, γ ) and ( X, γ ) be two generalized Minkowski spaces overthe same vector space X . The following statements are equivalent.(i) The gauges γ and γ generate the same topology, i.e., we have τ γ = τ γ .(ii) For all ( x n ) ∞ n =0 ⊂ X , lim n →∞ x n = x w.r.t. γ if and only if lim n →∞ x n = x w.r.t. γ .(iii) The gauges γ and γ are equivalent , i.e., there exist numbers c , c > suchthat c γ ( x ) ≤ γ ( x ) ≤ c γ ( x ) for all x ∈ X .Proof. First assume that (i) holds true. As B γ (0 , is a τ γ -neighborhood of , it is also a τ γ -neighborhood of , so there exists a number λ > such that THOMAS JAHN AND CHRISTIAN RICHTER B γ (0 , λ ) ⊂ B γ (0 , . Thus, for x = 0 , we have λγ ( x ) x ∈ B γ (0 , λ ) ⊂ B γ (0 , ,i.e., λγ ( x ) ≤ γ ( x ) . This means we can take c = λ in (iii). Interchanging theroles of γ and γ gives a similar expression for c in (iii).For (iii) ⇒ (ii), simply note that (iii) yields the equivalence of lim n →∞ γ ( x n − x ) = 0 with lim n →∞ γ ( x n − x ) = 0 .The implication (ii) ⇒ (i) is obvious, since closed sets are characterized by closednessunder limits of sequences. (cid:3) The following result gives a characterization when the topology generated by agauge can be generated by a norm and when it is a vector space topology, see also[4] for a related discussion in a broader setting.
Proposition 4.2.
Let ( X, γ ) be a generalized Minkowski space. The followingstatements are equivalent.(i) The topology τ γ is a vector space topology, i.e., the mappings X × X → X , ( x, y ) x + y and R × X → X , ( λ, x ) λx are continuous w.r.t. the producttopologies.(ii) The mapping X → X , x
7→ − x , is continuous at w.r.t. τ γ .(iii) The gauges γ and γ ∨ are equivalent.(iv) The gauge γ and the norm k x k γ = max { γ, γ ∨ } are equivalent.(v) There exists a norm k·k : X → R such that γ and k·k are equivalent.(vi) There exists a norm k·k : X → R such that τ γ = τ k·k .Proof. The implication (i) ⇒ (ii) is trivial. For (ii) ⇒ (iii), according to the assump-tion there exists λ > such that B γ (0 , λ ) is mapped into the neighborhood B γ (0 , of − . That is, − B γ (0 , λ ) ⊂ B γ (0 , , whence also B γ (0 , λ ) ⊂ − B γ (0 , .These give B γ ∨ (0 , λ ) ⊂ B γ (0 , , B γ (0 , λ ) ⊂ B γ ∨ (0 , and in turn λγ ( x ) ≤ γ ∨ ( x ) ≤ λ γ ( x ) .Now let us show (iii) ⇒ (iv). By assumption, there exist numbers c , c > suchthat c γ ( x ) ≤ γ ∨ ( x ) ≤ c γ ( x ) . But then also γ ( x ) ≤ max { γ ( x ) , γ ∨ ( x ) } ≤ max { γ ( x ) , c γ ( x ) } = max { , c } γ ( x ) for arbitrary x ∈ X .The implication (iv) ⇒ (v) is again trivial, (v) ⇒ (vi) follows from Proposition 4.1,and (vi) ⇒ (i) is trivial, too. (cid:3) As a consequence of (iii), balls B γ ( x, λ ) with x ∈ X and λ > are closed in τ γ .Indeed, for n ∈ N , let x n ∈ B γ ( x, λ ) and assume that there exists x ∈ X suchthat lim n →∞ x n = x w.r.t. γ . Then, by (iii), also lim n →∞ x n = x w.r.t. γ ∨ and γ ( x − x ) ≤ γ ( x − x n ) + γ ( x n − x ) ≤ γ ∨ ( x n − x ) + λ n →∞ −→ λ . Hence x ∈ B γ ( x, λ ) .Note that for general gauges γ , B γ (0 , need not be closed in τ γ . This is stated in[6, Proposition 1.1.8.1], but there the concept of a gauge is more flexible, since itrequires x = 0 if and only if γ ( x ) = 0 and γ ( − x ) = 0 , whereas we suppose x = 0 if and only if γ ( x ) = 0 . Therefore we add the following example that meets ourstronger concept of a gauge. Example 4.3.
Let the space X = ℓ of all absolutely convergent sequences of realsbe equipped with the gauge γ ( ξ , ξ , . . . ) = max (cid:8) sup i ≥ | ξ i | , P ∞ i =1 ξ i (cid:9) . Then the OPROXIMINALITY OF LINEAR SUBSPACES IN GENERALIZED MINKOWSKI SPACES 7 sequence ( x n ) ∞ n =1 ⊂ ℓ defined by x n = (cid:18) n , . . . , n | {z } n times , , , . . . (cid:19) satisfies γ ( x n ) ≡ and lim n →∞ ( − x n ) = 0 . Moreover, B γ (0 , is not closed in τ γ . Proof.
The claims γ ( x n ) = 1 and lim n →∞ ( − x n ) = 0 are obvious.To see that B γ (0 , is not closed, we consider x = (cid:0) , , , , . . . (cid:1) ∈ ℓ . Then γ ( x ) = 2 , whence x / ∈ B γ (0 , . The sequence ( x − x n ) ∞ n =1 ⊂ ℓ satisfies ( x − x n ) ∞ n =1 ⊂ B γ (0 , , because γ ( x − x n ) = max (cid:26) sup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) − n (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) − n (cid:12)(cid:12)(cid:12)(cid:12) , . . . , (cid:12)(cid:12)(cid:12)(cid:12) n − − n (cid:12)(cid:12)(cid:12)(cid:12) , n (cid:27) , − (cid:27) = 1 , and lim n →∞ ( x − x n ) = x , since lim n →∞ γ (( x − x n ) − x ) = lim n →∞ γ ( − x n ) = 0 .Having found a sequence ( x − x n ) ∞ n =1 ⊂ B γ (0 , such that lim n →∞ ( x − x n ) = x / ∈ B γ (0 , , we see that B γ (0 , is not closed. (cid:3) Taking into account that a gauge can behave strangely compared with a norm, westress here that the local topological behaviors of gauges and norms agree in thefollowing sense.
Lemma 4.4.
Let ( X, γ ) be a generalized Minkowski space and let X be a finite-dimensional linear subspace of X .(i) The topology on X generated by γ agrees with the canonical (Euclidean)vector space topology on X .(ii) The subspace X is closed in τ γ .Proof. For (i), see [8, Theorem 9]. For (ii), consider a sequence ( x n ) ∞ n =1 ⊆ X suchthat lim n →∞ x n = x ∈ X . Application of (i) to the finite-dimensional subspace X = lin ( X ∪ { x } ) shows that x ∈ X . So X is closed. (cid:3) Claim (i) and Proposition 4.1 imply in particular that all gauges on a finite-dimensional linear space are equivalent and give rise to the same concepts of bound-edness and convergence.5.
Characterizing Hilbert spaces in terms of coproximinality
Here we want to show the following characterization of Hilbert spaces.
Theorem 5.1.
Let ( X, γ ) be a generalized Minkowski space of dimension dim( X ) ≥ . The following statements are equivalent.(i) The generalized Minkowski space ( X, γ ) is a Hilbert space, i.e., γ is a norminduced by an inner product and ( X, γ ) is complete w.r.t. that norm.(ii) Every closed -codimensional linear subspace K of X is coproximinal. We recall that the above is established in [7, Theorem 1] under the additionalassumptions that γ is a norm and that the normed space ( X, γ ) is complete.5.1. The necessity of an inner product structure.
The coproximinality ofall linear subspaces of a fixed dimension implies the coproximinality of all linearsubspaces of lower finite dimension.
Proposition 5.2.
Let ( X, γ ) be a generalized Minkowski space. If there exists afinite-dimensional linear subspace X ⊂ X which is not coproximinal, then therealso exists a closed -codimensional linear subspace H ⊂ X with X ⊂ H such that H as well as all subspaces X with X ⊂ X ⊂ H are not coproximinal. THOMAS JAHN AND CHRISTIAN RICHTER
Proof. As X is not coproximinal, there exists a point y ∈ X such that(5.1) Q X ( y ) = X ∩ \ z ∈ X B γ ( z, γ ( y − z )) | {z } =: C = ∅ . Then there exists some n ∈ { , , . . . } such that(5.2) X ∩ [ x ∈ C U γ ( x, n ) | {z } =: G ( n ) = ∅ . Indeed, suppose, contrary to (5.2), that there are x n ∈ X ∩ G ( n ) for n = 1 , , . . . We fix some z ∈ X and obtain ( x n ) ∞ n =1 ⊂ X ∩ G (1) ⊂ X ∩ [ x ∈ B γ ( z ,γ ( y − z )) U γ ( x, ⊂ X ∩ U γ ( z , γ ( y − z ) + 1) . So ( x n ) ∞ n =1 is bounded in the (closed) finite-dimensional subspace X (cf. Lemma 4.4),thus having a convergent subsequence. W.l.o.g., ( x n ) ∞ n =1 converges itself, i.e., lim n →∞ γ ( x n − x ) = 0 for some x ∈ X . By Lemma 4.4, also lim n →∞ γ ( x − x n ) =0 . Since x n ∈ G ( n ) , there exist ˜ x n ∈ C such that γ ( x n − ˜ x n ) < n . Then every z ∈ X satisfies γ ( x − z ) ≤ γ ( x − x n ) + γ ( x n − ˜ x n ) + γ (˜ x n − z ) < γ ( x − x n ) + 1 n + γ ( y − z ) for all n = 1 , , . . . , whence x ∈ B γ ( z, γ ( y − z )) . Therefore x ∈ C in addition to x ∈ X . This contradicts (5.1) and proves (5.2).Now Theorem 2.2.8 from [6] says that there is a continuous linear functional h from ( X, γ ) into R that separates the open convex set G ( n ) from the disjoint linearsubspace X in the sense that(5.3) h ( x ) < h ( z ) = 0 for all x ∈ G ( n ) , z ∈ X . The null space H := h − (0) is a closed -codimensional subspace of X .Let X be a linear subspace of X such that X ⊂ X ⊂ H . Then we have Q X ( y ) = X ∩ \ z ∈ X B γ ( z, γ ( y − z )) ⊂ H ∩ \ z ∈ X B γ ( z, γ ( y − z ))= H ∩ C (5.2) ⊂ H ∩ G ( n ) (5.3) = ∅ , which means that X is not coproximinal. (cid:3) Corollary 5.3.
Let ( X, γ ) be a generalized Minkowski space of dimension dim( X ) ≥ . If every closed -codimensional linear subspace K of X is coproximinal, then γ is a norm induced by an inner product on X .Proof. By [14, Section 4], it is sufficient to show that the restriction of γ to every -dimensional linear subspace of X is a norm induced by an inner product. To thisend, fix an arbitrary -dimensional subspace X of X . By Proposition 5.2, all -dimensional or -dimensional linear subspaces of X are coproximinal. In particular,all -dimensional or -dimensional linear subspaces of X are coproximinal, alsowhen viewed in ( X , γ | X ) . By Theorem 3.2, the generalized Minkowski space ( X , γ | X ) is in fact a normed space (of dimension ), and since each of its -dimensional linear subspaces is coproximinal, [7, Theorem 1] shows that γ | X is a OPROXIMINALITY OF LINEAR SUBSPACES IN GENERALIZED MINKOWSKI SPACES 9 norm induced by an inner product. As every -dimensional linear subspace of X is a linear subspace of some choice of X , we have shown that the restriction of γ to every -dimensional linear subspace of X is in fact a norm induced by an innerproduct. (cid:3) The necessity of completeness.
First we note that in the setting of aninner product space points of best coapproximation within a linear subspace arefound by the orthogonal projection onto that space, see also [7, Section 2].
Lemma 5.4.
Let K be a linear subspace of an inner product space ( X, h· | ·i ) , let y ∈ X and let z ∈ K . Then z ∈ Q K ( y ) if and only if y − z ⊥ K , i.e., h y − z | z i = 0 for all z ∈ K .Proof. We reformulate the claim z ∈ Q K ( y ) successively: ⇔ k z − ( z + λz ) k ≤ k y − ( z + λz ) k for all z ∈ K, λ ∈ R ⇔ k λz k ≤ k ( y − z ) − λz k for all z ∈ K, λ ∈ R ⇔ k λz k ≤ k y − z k − h y − z | λz i + k λz k for all z ∈ K, λ ∈ R ⇔ λ h y − z | z i ≤ k y − z k for all z ∈ K, λ ∈ R ⇔ h y − z | z i = 0 for all z ∈ K (cid:3) Now the necessity of completeness is obtained via the Riesz representation theoremof continuous linear functionals.
Lemma 5.5.
The following are equivalent for every inner product space ( X, h· | ·i ) .(i) The vector space X is complete w.r.t. the norm k·k induced by h· | ·i .(ii) For every continuous linear functional f : X → R , there exists x ∈ X suchthat f ( x ) = h x | x i for all x ∈ X (Riesz representation theorem).(iii) Every closed -codimensional linear subspace K of X is coproximinal.Proof. For (i) ⇔ (ii), see [10, Theorem 3.3.5]. For (i) ⇒ (iii), we use the fact that theorthogonal projection P K : X → K is well-defined by (i) (cf. [10, Theorem 3.3.5and Definition 3.3.9]), and that P K maps every y ∈ X onto P K ( y ) ∈ Q K ( y ) byLemma 5.4.For (iii) ⇒ (ii), let f : X → R be a continuous linear functional. We can assume that f = 0 . Then the null space K := f − (0) is a closed -codimensional subspace of X .We pick y ∈ X \ K . By (iii), there is z ∈ Q K ( y ) . Lemma 5.4 yields y − z ⊥ K .We put x := f ( y ) k y − z k ( y − z ) . For every x ∈ X , there exist z ∈ K and λ ∈ R such that x = z + λ ( y − z ) , since X = K ⊕ lin ( { y − z } ) . Thus f ( x ) = f ( z + λ ( y − z ))= f ( z − λz ) + λf ( y )= 0 + * λ ( y − z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( y ) k y − z k ( y − z ) + , since z, z ∈ K = f − (0) , = h z | x i + h λ ( y − z ) | x i , since x ⊥ K, = h x | x i . (cid:3) Conclusion.
The implication (i) ⇒ (ii) from Theorem 5.1 can be found in [7,Theorem 1]. (In fact, using the reformulation from Lemma 5.4 it is folklore.) For(ii) ⇒ (i), we combine Corollary 5.3 with Lemma 5.5. Acknowledgements.
T.J. would like to acknowledge support by the DFG Ul-403/2-1 and NUTRICON project.
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Email address : [email protected] (C. Richter) Institute for Mathematics, Friedrich Schiller University, 07737 Jena,Germany
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