in fact we have equality. We have {is, s ∈ S} is a family of linear operators bounded on X∗, since for all `, sups∈S ix(`) = sups∈S `(s) which is finite by assumption. Therefore by the Uniform Boundedness Principle (X ∗ is a Banach space), {is, s ∈ S} are uniformly bounded, i.e. there exists M such that for all s ∈ S, ‖is‖X∗∗ ≤ M , i.e. ‖s‖X ≤M . (11.3) Let X,Y be Banach spaces. Let L ∈ B(X,Y )...

For two normed algebras $A$ and $B$ with the character space   $bigtriangleup(B)neq emptyset$  and a left $B-$module $X,$  a certain class of bounded linear maps from $A$ into $X$ is introduced. We set $CMH_B(A, X)$  as the set of all non-zero $B-$character module homomorphisms from $A$ into $X$. In the case where $bigtriangleup(B)=lbrace varphirbrace$ then $CMH_B(A, X)bigcup lbrace 0rbrace$ is...

where En are measurable sets. Let > 0. Since μ is outer regular, and inner regular on measurable sets of finite measure, there exist an open set On and a compact set Kn such that Kn ⊂ En ⊂ On μ(En)− 1 2N < μ(Kn) ≤ μ(En) ≤ μ(On) < μ(En) + 1 2N thus Kn ⊂ On and μ(On)− μ(Kn) < 1 N , and since X is of finite measure, we have μ(On \Kn) < 1 N consider the closed sets Kn and O c n. By Urysohn’s Lemma,...

(4.1) Let P : R → R be a polynomial which does not vanish identically. Use Fubini’s Theorem to show, by induction on the dimension n, that Z(P ) = {x ∈ R|P (x) = 0} has Lebesgue measure equal to zero. proof By induction on n. For n = 1, consider a polynomial P ∈ R[X], and let d = degP . Then P has at most d roots, and Z(P ) = {x ∈ R|P (x) = 0} is finite, thus has measure 0. Now let n ≥ 1, and a...

we have by construction μ(En) = an. An important observation is that a subset of consecutive intervals En, En+1, . . . , En+k are either pairwise disjoint, or they cover the whole interval [0, 1]. Now for all n, let fn = 1En . Then we have • ‖fn‖1 = ∫ X |1En |dμ = μ(En) = an. • for all x ∈ [0, 1], (fn(x))n does not converge. Indeed, if we suppose by contradiction that (fn(x))n converges for som...

This paper is primarily concerned with the meet structure of the lattice of type Bn partitions of the set {±1, ±2, . . . , ±n}. The lattice of type Bn set partitions has been studied by Reiner [8]. It can be regarded as a representation of the intersection lattice of the type B Coxeter arrangements, see Björner and Wachs [3], Björner and Brenti [2] and Humphreys [6]. A set partition of type Bn ...

In this paper, we have dealt with the concept of fuzzy b-closed and fuzzy bcompact set, and we study the characterization of fuzzy b-compact set by using fuzzy filterbases. We consider some relations between the fuzzy b-compact space and fuzzy b-closed space. Finally, the concept of bounded and semi bounded operators between fuzzy normed space are studied and some immediate results are also pro...

(10.1) Let k ∈ N, and let X = C k ([0, 1] d) to be the set of all functions f ∈ C 0 ([0, 1] d) such that for all α ∈ ∆ k , f (α) = ∂f ∂x α 1 1 ...∂x α d d exists and is continuous on (0, 1) d , and extends continuously on [0, 1] d. Here

This appendix contains the results obtained from the analysis of variance described in chapter three using " set " as the second factor.