Аннотация:
Let
$(A,\mathscr{A},\mu)$
be a
$\sigma$-finite complete measure space, and let
$p(\cdot)$
be a
$\mu$-measurable function on
$A$
which takes values in
$(1,\infty)$.
Let
$Y$
be a
subspace of a Banach space
$X$.
By
$\widetilde{L}^{p(\cdot),\varphi}(A, Y)$
and
$\widetilde{L}^{p(\cdot),\varphi}(A, X)$
we denote the grand Bochner–Lebesgue spaces
with variable
exponent
$p(\cdot)$
whose functions take values in
$Y$
and
$X$,
respectively.
First, we
estimate the distance of
$f$
from
$\widetilde{L}^{p(\cdot),\varphi}(A, Y)$
when
$f\in
\widetilde{L}^{p(\cdot),\varphi}(A, X)$.
Then we prove that
$\widetilde{L}^{p(\cdot),\varphi}(A, Y)$
is proximinal in
$\widetilde{L}^{p(\cdot),\varphi}(A, X)$
if
$Y$
is weakly
$\mathcal{K}$-analytic and
proximinal in
$X$.
Finally, we establish a connection between the proximinality of
$\widetilde{L}^{p(\cdot),\varphi}(A, Y)$
in
$\widetilde{L}^{p(\cdot),\varphi}(A, X)$
and the proximinality of
$L^1(A, Y)$
in
$L^1(A, X)$.
Ключевые слова:proximinality, grand Bochner–Lebesgue space, variable exponent, best approximation,
weakly
$\mathcal{K}$-analytic.
The research of the second author was supported by the Natural Science Foundation of Hainan Province (Grant No. 2018CXTD338)
and the National Natural Science Foundation of China (Grant No. 11761026 and 11761027).
\Bibitem{WeiXu19}
\by Haihua~Wei, Jingshi~Xu
\paper Proximinality in Banach space valued grand Bochner-Lebesgue spaces with variable exponent
\jour Math. Notes
\yr 2019
\vol 105
\issue 4
\pages 618--624
\mathnet{http://mi.mathnet.ru/mzm11273}
\crossref{https://doi.org/10.1134/S0001434619030349}
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Эта публикация цитируется в следующих 1 статьяx:
P. Jain, M. Singh, A. Singh, V. D. Stepanov, “On the Duality of Grand Bochner–Lebesgue Spaces”, Матем. заметки, 107:2 (2020), 247–256; P. Jain, M. Singh, A. Singh, V. D. Stepanov, “On the Duality of Grand Bochner–Lebesgue Spaces”, Math. Notes, 107:2 (2020), 247–256