• Korean Journal of Mathematics
• /
• v.15 no.1
• /
• pp.13-26
• /
• 2007

We prove that every strong null sequence in a Banach space X lies inside the range of a vector measure of bounded variation if and only if the condition $\mathcal{N}_1(X,{\ell}_1)={\Pi}_1(X,{\ell}_1)$ holds. We also prove that for $1{\leq}p<{\infty}$ every strong ${\ell}_p$ sequence in a Banach space X lies inside the range of an X-valued measure of bounded variation if and only if the identity operator of the dual Banach space $X^*$ is ($p^{\prime}$,1)-summing, where $p^{\prime}$ is the conjugate exponent of $p$. Finally we prove that a Banach space X has the property that any sequence lying in the range of an X-valued measure actually lies in the range of a vector measure of bounded variation if and only if the condition ${\Pi}_1(X,{\ell}_1)={\Pi}_2(X,{\ell}_1)$ holds.

• Bulletin of the Korean Mathematical Society
• /
• v.43 no.3
• /
• pp.551-558
• /
• 2006

We establish a weak law of large numbers for sequence of random elements with values in p-uniformly smooth Banach space. Our result is more general and stronger than some well-known ones.

• Journal of the Korean Mathematical Society
• /
• v.41 no.1
• /
• pp.157-174
• /
• 2004

In this work we discuss "plank problems" for complex Banach spaces and in particular for the classical $L^{p}(\mu)$ spaces. In the case $1\;{\leq}\;p\;{\leq}\;2$ we obtain optimal results and for finite dimensional complex Banach spaces, in a special case, we have improved an early result by K. Ball [3]. By using these results, in some cases we are able to find best possible lower bounds for the norms of homogeneous polynomials which are products of linear forms. In particular, we give an estimate in the case of a real Hilbert space which seems to be a difficult problem. We have also obtained some results on the so-called n-th (linear) polarization constant of a Banach space which is an isometric property of the space. Finally, known polynomial inequalities have been derived as simple consequences of various results related to plank problems.

• Communications of the Korean Mathematical Society
• /
• v.22 no.2
• /
• pp.209-218
• /
• 2007

In this paper we investigate the ${\ell}^p$ space structure of the Banach envelope of $WeakL_1$. In particular, the Banach envelope of $WeakL_1$ contains a complemented Banach sublattice that is isometrically isomorphic to the nonseparable Banach lattice ${\ell}^p$, ($1{\leq}p<\infty$) as well as the separable case.

• Journal of the Chungcheong Mathematical Society
• /
• v.13 no.1
• /
• pp.71-78
• /
• 2000

The main goal of this paper is to prove the following theorem ; Let (X, ${\rho}_1$) be a fuzzy normed linear space over K and (Y, ${\rho}_2$) be a fuzzy Banach space over K. If ${\chi}_{B_{{\parallel}{\cdot}{\parallel}}}{\supseteq}{\rho}*$, then (CF(X,Y), ${\rho}*$) is a fuzzy Banach space, where ${\rho}*(f)={\vee}{\lbrace}{\theta}{\wedge}\frac{1}{t({\theta},f)}\;{\mid}\;{\theta}{\in}(0,1){\rbrace}$, $f{\in}CF(X,Y)$, $B_{{\parallel}{\cdot}{\parallel}}$ is the closed unit ball on (CF(X, Y), ${\parallel}{\cdot}{\parallel}$ and ${\parallel}f{\parallel}={\vee}{\lbrace}P^2_{{\alpha}^-}(f(x))\;{\mid}\;P^1_{{\alpha}^-}(x)=1,\;x{\in}X{\rbrace}$, $f{\in}CF(X,Y)$, ${\alpha}{\in}(0,1)$.

• Korean Journal of Mathematics
• /
• v.9 no.2
• /
• pp.83-98
• /
• 2001

We characterize ($r,w$)-nuclear operators acting from a Banach space whose dual has weak type $q$ into a Banach space of weak type $p$ by the asymptotic behaviour of their entropy numbers.

• Communications of the Korean Mathematical Society
• /
• v.18 no.3
• /
• pp.491-499
• /
• 2003

The Banach envelope of $WeakL_1$ contains a complemented Banach sublattice that is isometrically isomorphic to Lp($\mu$) space.

• Journal of the Chungcheong Mathematical Society
• /
• v.19 no.3
• /
• pp.263-274
• /
• 2006

In this paper, we prove the Hyers-Ulam-Rassias stability of the quadratic mapping in generalized quasi-Banach spaces, and of the quadratic mapping in generalized p-Banach spaces.

• Journal of the Korean Mathematical Society
• /
• v.32 no.4
• /
• pp.679-688
• /
• 1995

Let $(B, \left\$\mid$ \right\$\mid$)$ be a real separable Banach space. Let $(\Omega, F, P)$ denote a probability space. A random elements in B is a function from $\Omega$ into B which is $F$-measurable with respect to the Borel $\sigma$-field $B$(B) in B.

• Korean Journal of Mathematics
• /
• v.32 no.3
• /
• pp.545-560
• /
• 2024

In the present study, we have constructed a new Banach series space |𝛶^r|^u_p by using concept of absolute Jordan totient summability |𝛶^r, u_n|_p which is derived by the infinite regular matrix of the Jordan's totient function. Also, we prove that the series space |𝛶^r|^u_p is linearly isomorphic to the space of all p-absolutely summable sequences ℓ_p for p ≥ 1. Moreover, we compute the α-, β- and γ- duals of this space and construct Schauder basis for the series space |𝛶^r|^u_p. Finally, we characterize the classes of infinite matrices (|𝛶^r|^u_p, X) and (X, |𝛶^r|^u_p), where X is any given classical sequence spaces ℓ_∞, c, c₀ and ℓ₁.