Inequalities Involving Lower-Triangular Matrices

Abstract

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 .

Best possible results in a class of inequalities

“…Suppose 1 < p ^ 00, and rp > 1 if p < 00, γ > 0 if Proof. It has been mentioned earlier that {n r~1 } e I (see [3], §8). Suppose be I, and b < {n r~1 }.…”

“…It was shown in [3,Theorem 9.3] that the inequality (2) with b = {n r~1 } is not best possible, and that, indeed, there is no best possible inequality (2) for this triple A, λ, μ. (It is interesting to note that if p = 1, r > 1, the inequality (2) with 6 = {n^1} holds and is best possible (by [3, Corollary 4.9]).)…”

“…Hardy [2, Theorem 215] proved the following inequality: In [3] we had tried to systematize and unify inequality results of the following form:…”

“…In [3] we ordered the sequences in w by defining that for 6, cew 9 b < c (b is less than c) if and only if, for some M > 0, \b n \ ^M\c n \ for all n. Now one can observe that an inequality of the form (2) is better the smaller the K, or the larger the sequence b with the notion of largeness of sequences defined above.…”

Generalizations of Hardy's and Copson's Inequalities

“…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.…”