On an Inequality of Hardy's (Ii)

A subset A of o> is normal if x e A implies N(x) c A. If A <= to, a subset fi of A is said to be normal in A if and only if x e ft implies N(x)nX <= p. Clearly, if A is normal, normal in A means just normal, for subsets of A.If A is a subspace of o>, a normal topology on A is a t.v.s. topology on A for which there exists a neighbourhood base at the origin consisting of sets which are normal in A.If A is a normal subspace of co, and T is a normal topology on A, then < •, e n > is continuous on (A, T) for all n e N if and only if (A, T) is a Hausdorff space (see [11, Proposition 2.7]). Thus a normal subspace of oo with a complete metric, normal topology is an FK space.If (E, T) is a t.v.s., a function p from E into the non-negative reals will be said to determine the topology T if the sets {x e E: p(x) < e}, with e > 0, comprise a neighbourhood base at the origin in {E, T). If A is a subspace of oo, a function p from A into the non-negative reals will be called normal if and only if a; 6 A and y e XnN(x) imply p(y) ^ p(x). If the normal function p on A determines a topology T on A, then T is a normal topology. The topology of a Hausdorff t.v.s. is determined by some function p if and only if the space is metrizable, in which case p may be replaced, if desired, by a pseudonorm [20, Theorem 6.1, p. 28]. In this paper the topology-determining functions will be norms or quasinorms, although some of the more general considerations are applicable to spaces with more disturbing topology-determining functions. (We are thinking of the spaces l{p n ), inequalities related to which will not be studied here; see [15] and [22].) A quasinorm || • || on a vector space!? is like a norm except that, instead of the triangle inequality, || • || satisfies || x+y \\ < M(\\ x \\ +1| y \\) for all x,y e E, for some M > 1. {li M = 1,||-|| is a norm.) If (^,||-b), (F, || • H^) are quasinormed spaces, a linear map T from E to F is continuous From Proposition 2.2 and the remarks of §1, we have the following corollary. COROLLARY 2.3. Suppose (ytx, ||*|| A ) and (A, ||-|| A ) are quasinormed FK spaces, and suppose X and ||*|| A are normal. Suppose /z satisfies (9). Then D(fj,, nor-A~1(X)) is a normal FK space, with the normal quasinorm \\-\\ defined by \\b\\ = s u p m^\ \ A \ b x \ | | A .

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“…REMARKS, (a) For 1 < p < oo, and z v ...,z n not all zero, the inequality (23) is strict, by the proof in [4].…”

Section: Two Results Of Petersen and Davies And Petersenmentioning

confidence: 95%

“…Petersen [18] and subsequently Davies and Petersen [4] produced sufficient conditions on a matrix A and an auxiliary sequence/ £ co for the existence of an inequality of the form \\A\x\\\ p^K \\f.a.x\\ p (l^p<co) for some K with a = {a mm }, the main diagonal sequence of A.…”

Section: Introductionmentioning

confidence: 99%

Inequalities Involving Lower-Triangular Matrices

Johnson

Mohapatra

1980

Proceedings of the London Mathematical Society

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“…By using the following inequality (see [1,7] Dividing both sides of (2.6) by (m α + n α ) 1/α , and then taking the sum over n from 1 to r first and then the sum over m from 1 to k and using the Schwarz inequality and then interchanging the order of the summations (see [7,8]) we observe that…”

Section: Resultsmentioning

confidence: 99%

“…These two theorems were studied extensively and numerous variants, generalizations, and extensions appeared in the literature, see [1,2,3,5,6,9] and the references therein.…”

Section: )mentioning

confidence: 99%

An improvement of some inequalities similar to Hilbert′s inequality

Kim

2001

International Journal of Mathematics and Mathematical Sciences

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“…Copson [ In this paper I replace these means by more general linear transforms, in both (1) and the companion inequality; this is done in Theorems 1, 2 and 3. In Theorems 4, 5 and 6,1 further replace the index p by a different index q on the left (but not on the right), obtaining inequalities of the form ||Ax||^C||x|| p , (2) where A is an infinite square matrix and x is a variable column, and the norms are weighted versions of the norms in l q and P. The matrix A is sometimes (but not always) lower or upper triangular, as in the work of Davies and Petersen [3], Johnson and Mohapatra [5], and Redheffer [7]. The words increasing and decreasing are used throughout in preference to nondecreasing and non-increasing.…”

Section: Introductionmentioning

confidence: 99%

Generalizations of Hardy's and Copson's Inequalities

Love

1984

Journal of the London Mathematical Society

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On an Inequality of Hardy's (Ii)

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Inequalities Involving Lower-Triangular Matrices

Inequalities Involving Lower-Triangular Matrices

An improvement of some inequalities similar to Hilbert′s inequality

Generalizations of Hardy's and Copson's Inequalities

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