scholarly journals A distance function property implying differentiability

1989 ◽  
Vol 39 (1) ◽  
pp. 59-70 ◽  
Author(s):  
J.R. Giles
Keyword(s):  
Linear Space ◽  
Distance Function ◽  
Normed Linear Space ◽  
Closed Set ◽  
Differentiable Norm ◽  
Function Property ◽  
Fréchet Differentiable ◽  
Image Position ◽  
Real Normed Linear Space

In a real normed linear space X, properties of a non-empty closed set K are closely related to those of the distance function d which it generates. If X has a uniformly Gâteaux (uniformly Fréchet) differentiable norm, then d is Gâteaux (Fréchet) differentiable at x ∈ X/K if there exists an such thatand is Géteaux (Fréchet) differentiable on X / K if there exists a set P+(K) dense in X/K where such a limit is approached uniformly for all x ∈ P+(K). When X is complete this last property implies that K is convex.

scholarly journals Operators with a Single Spectrum

1966 ◽  
Vol 15 (1) ◽  
pp. 11-18 ◽  
Author(s):  
T. T. West
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Linear Operators ◽  
Complex Field ◽  
Unique Extension ◽  
Single Spectrum ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Resolvent Set ◽  
Image Position

Let X be an infinite dimensional normed linear space over the complex field Z. X will not be complete, in general, and its completion will be denoted by . If ℬ(X) is the algebra of all bounded linear operators in X then T ∈ ℬ(X) has a unique extension and . The resolvent set of T ∈ ℬ(X) is defined to beand the spectrum of T is the complement of ρ(T) in Z.

scholarly journals Pythagorean Orthogonality in a Normed Linear Space

1958 ◽  
Vol 9 (4) ◽  
pp. 168-169
Author(s):  
Hazel Perfect
Keyword(s):  
Euclidean Space ◽  
Linear Space ◽  
Normed Linear Space ◽  
Image Position ◽  
Pythagorean Orthogonality

This note presents a proof of the following proposition:Theorem. If Pythagorean orthogonality is homogeneous in a normed linear space T then T is an abstract Euclidean space.The theorem was originally stated and proved by R. C. James ([1], Theorem 5. 2) who systematically discusses various characterisations of a Euclidean space in terms of concepts of orthogonality. I came across the result independently and the proof which I constructed is a simplified version of that of James. The hypothesis of the theorem may be stated in the form:Since a normed linear space is known to be Euclidean if the parallelogram law:is valid throughout the space (see [2]), it is evidently sufficient to show that (l) implies (2).

scholarly journals On Exposed and Farthest Points in Normed Linear Spaces

1971 ◽  
Vol 12 (3) ◽  
pp. 301-308 ◽  
Author(s):  
M. Edelstein ◽  
J. E. Lewis
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Nonempty Subset ◽  
Linear Spaces ◽  
Farthest Points ◽  
Normed Linear Spaces ◽  
Farthest Point ◽  
Image Position

Let S be a nonempty subset of a normed linear space E. A point s0 of S is called a farthest point if for some x ∈ E, . The set of all farthest points of S will be denoted far (S). If S is compact, the continuity of distance from a point x of E implies that far (S) is nonempty.

scholarly journals On Uniqueness of New Orthogonality via 2-HH Norm in Normed Linear Space

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Bhuwan Prasad Ojha ◽  
Prakash Muni Bajracharya ◽  
Vishnu Narayan Mishra
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Inner Product ◽  
Isosceles Orthogonality ◽  
Complete Proof ◽  
Underlying Space ◽  
Real Inner Product Space ◽  
Special Case ◽  
Real Normed Linear Space ◽  
Pythagorean Orthogonality

This paper generalizes the special case of the Carlsson orthogonality in terms of the 2-HH norm in real normed linear space. Dragomir and Kikianty (2010) proved in their paper that the Pythagorean orthogonality is unique in any normed linear space, and isosceles orthogonality is unique if and only if the space is strictly convex. This paper deals with the complete proof of the uniqueness of the new orthogonality through the medium of the 2-HH norm. We also proved that the Birkhoff and Robert orthogonality via the 2-HH norm are equivalent, whenever the underlying space is a real inner-product space.

Linear Normed Spaces with Extension Property

1966 ◽  
Vol 9 (4) ◽  
pp. 433-441 ◽  
Author(s):  
George Elliott ◽  
Israel Halperin
Keyword(s):  
Linear Space ◽  
Linear Extension ◽  
Normed Linear Space ◽  
Extension Property ◽  
Linear Mapping ◽  
Normed Spaces ◽  
Bounded Linear ◽  
The Real ◽  
Real Normed Linear Space

In this paper we shall say “E has the (F, G) (extension) property” to mean the following: F is a subspace of the real normed linear space G, E is a real normed linear space, and any bounded linear mapping F→E has a linear extension G→E with the same bound (equivalently, every linear mapping F→E of bound 1 has a linear extension G→E with bound 1).

scholarly journals Summable Family in a Commutative Group

2015 ◽  
Vol 23 (4) ◽  
pp. 279-288
Author(s):  
Roland Coghetto
Keyword(s):  
Topological Space ◽  
Linear Space ◽  
Metric Space ◽  
Normed Linear Space ◽  
Linear Topological Space ◽  
Commutative Group ◽  
Image Filter ◽  
Directed Set ◽  
Generic Theory ◽  
Real Normed Linear Space

Summary Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [22], [7]. In this paper we present our formalization of this theory in Mizar [6]. First, we compare the notions of the limit of a family indexed by a directed set, or a sequence, in a metric space [30], a real normed linear space [29] and a linear topological space [14] with the concept of the limit of an image filter [16]. Then, following Bourbaki [9], [10] (TG.III, §5.1 Familles sommables dans un groupe commutatif), we conclude by defining the summable families in a commutative group (“additive notation” in [17]), using the notion of filters.

Minimal projections in Lp-spaces

1985 ◽  
Vol 97 (1) ◽  
pp. 127-136 ◽  
Author(s):  
E. J. Halton ◽  
W. A. Light
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Minimal Projection ◽  
Proper Subspace ◽  
Linear Map ◽  
Lp Spaces ◽  
Minimal Projections ◽  
Image Position

Let X be a normed linear space and let W be a proper subspace of X. A projection is a surjective linear map P: X → W such that P is idempotent. It is immediately clear that P has norm at least unity. Thus the problem of calculating the numberhas some interest. The number λ(W, X) is often called the relative projectiion constant of W in X. If the infimum is attained, any attaining projection is called a minimal projection. The problems of calculating λ(W, X) for a fixed X and W or finding a minimal projection turn out to be very dificult. For example, if X = C [0, 1] with the usual supremem norm and W is the subspace of polynominals of degree at most two then λ(W, X) remains unknown as does any example of a minimal projection. One of the few places where the problem shows much tractability is the case

A Note on Best Simultaneous Approximation in Normed Linear Spaces

1976 ◽  
Vol 19 (3) ◽  
pp. 359-360 ◽  
Author(s):  
Arne Brøndsted
Keyword(s):  
Linear Space ◽  
Convex Subset ◽  
Convex Functions ◽  
Existence And Uniqueness ◽  
Simultaneous Approximation ◽  
Normed Linear Space ◽  
Present Note ◽  
Linear Spaces ◽  
Best Simultaneous Approximation ◽  
Image Position

The purpose of the present note is to point out that the results of D. S. Goel, A. S. B. Holland, C. Nasim and B. N. Sahney [1] on best simultaneous approximation are easy consequences of simple facts about convex functions. Given a normed linear space X, a convex subset K of X, and points x1, x2 in X, [1] discusses existence and uniqueness of K* ∈ K such that

The approximation of bivariate functions by sums of univariate ones using the L1-metric

1982 ◽  
Vol 25 (2) ◽  
pp. 173-181 ◽  
Author(s):  
W. A. Light ◽  
J. H. McCabe ◽  
G. M. Phillips ◽  
E. W. Cheney
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Approximation Problem ◽  
The One ◽  
Image Position ◽  
Special Case ◽  
Bivariate Functions

We shall study a special case of the following abstract approximation problem: givena normed linear space E and two subspaces, M1 and M2, of E, we seek to approximate f ∈ E by elements in the sum of M1 and M2. In particular, we might ask whether closest points to f from M = M1 + M2 exist, and if so, how they are characterised. If we can define proximity maps p1 and p2 for M1 and M2, respectively, then an algorithm analogous to the one given by Diliberto and Straus [4] can be defined by the formulae

Best Possible Nets in a Normed Linear Space1

1975 ◽  
Vol 18 (1) ◽  
pp. 45-48 ◽  
Author(s):  
L. L. Keener
Keyword(s):  
Linear Space ◽  
Normed Linear Space ◽  
Sufficient Condition ◽  
Bounded Set ◽  
Image Position

In this note we examine the question of the existence of a best possible N-net for a bounded set in a normed linear space. A sufficient condition for existence is given which leads to easy proofs of some of the standard results. The pertinent reference here is the paper by Garkavi [1].Let E be a normed linear space and let M be a bounded set in E. Any system of N points in E will be called an N-net. For a given M and the net SN = {y1, y2,…, yN} defineand

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