• Bulletin of the Korean Mathematical Society
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• v.36 no.3
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• pp.589-597
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• 1999

Let G be a closed subspace of a Banach space X and let (S,$\Omega$,$\mu$) be a $\sigma$-finite measure space. It was known that $L_1$(S,G) is proximinal in $L_1$(S,X) if and only if $L_p$(S,G) is proximinal in $L_p$(S,X) for 1$\infty$. In this article we show that this result remains true when "proximinal" is replaced by "Chebyshev". In addition, it is shown that if G is a proximinal subspace of X such that either G or the kernel of the metric projection $P_G$ is separable then, for 0 < p $\leq$ $\infty$. $L_p$(S,G) is proximinal in $L_p$(S,X)

• Bulletin of the Korean Mathematical Society
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• v.38 no.1
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• pp.65-69
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• 2001

For any closed subspace X of $\ell_p, \; 1<\kappa<\infty$, K(X) is proximinal in L(X), and if X is a Banach space with an unconditional shrinking basis, then K(X, c$_0$) is proximinal in L(X,$ \ell_\infty$).

• The Mathematical Education
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• v.23 no.1
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• pp.67-68
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• 1984

• Journal of the Korean Mathematical Society
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• v.32 no.3
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• pp.415-425
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• 1995

• Bulletin of the Korean Mathematical Society
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• v.33 no.3
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• pp.427-434
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• 1996

The theory of numerical ranges in unital normed algebras has extensively studied by many authors, for examples, see [1], [2] for details.

• Bulletin of the Korean Mathematical Society
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• v.43 no.2
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• pp.377-387
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• 2006

In this paper, some characterizations of representation for continuous linear functionals on $2{\kappa}$-inner product spaces in terms of best approximations and orthogonalities are given.

• East Asian mathematical journal
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• v.16 no.2
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• pp.205-214
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• 2000

In this paper, we shall give new characterizations of best approximation element in linear 2-normed spaces in terms of bounded linear 2-functionals and 2-hyperplanes.

• Bulletin of the Korean Mathematical Society
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• v.40 no.3
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• pp.503-508
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• 2003

We prove that every M-ideal is strongly proximinal and that, for any Banach space X, $K(X,\;c_0)$ is an M-ideal in $L(X,\;\ell^{\infty})$.

• Bulletin of the Korean Mathematical Society
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• v.45 no.3
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• pp.601-606
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• 2008

The purpose of this paper is to introduce and discuss the concept of ${\omega}$-Chebyshev subspaces in Banach spaces. The concept of quasi Chebyshev in Banach space is defined. We show that ${\omega}$-Chebyshevity of subspaces are a new class in approximation theory. In this paper, also we consider orthogonality in normed spaces.

• The Pure and Applied Mathematics
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• v.8 no.2
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• pp.153-162
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• 2001

In this Paper, the notion of $\varepsilon$-Birkhoff orthogonality introduced by Dragomir [An. Univ. Timisoara Ser. Stiint. Mat. 29(1991), no. 1, 51-58] in normed linear spaces has been extended to metric linear spaces and a decomposition theorem has been proved. Some results of Kainen, Kurkova and Vogt [J. Approx. Theory 105 (2000), no. 2, 252-262] proved on e-near best approximation in normed linear spaces have also been extended to metric linear spaces. It is shown that if (X, d) is a convex metric linear space which is pseudo strictly convex and M a boundedly compact closed subset of X such that for each $\varepsilon$>0 there exists a continuous $\varepsilon$-near best approximation $\phi$ : X → M of X by M then M is a chebyshev set .