|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'500'096
Articles rated: 2609
19 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
Uniqueness of Hahn-Banach extension and related norm-$1$ projections in dual spaces | | Soumitra Daptari
; Tanmoy Paul
; T.S.S.R.K. Rao
; | | Date: |
21 Sep 2020 | | Abstract: | In this paper we study two properties viz. property-$U$ and property-$SU$ of
a subspace $Y$ of a Banach space which correspond to the uniqueness of the
Hahn-Banach extension of each linear functional in $Y^*$ and in addition to
that this association forms a linear operator of norm-1 from $Y^*$ to $X^*$. It
is proved that, under certain geometric assumptions on $X, Y, Z$ these
properties are stable with respect to the injective tensor product; $Y$ has
property-$U$ ($SU$) in $Z$ if and only if $Xotimes_e^vee Y$ has property-$U$
($SU$) in $Xotimes_e^vee Z$. We prove that when $X^*$ has the
Radon-Nikod$acute{y}$m Property for $1<p< infty$, $L_p(mu, Y)$ has
property-$U$ (property-$SU$) in $L_p(mu, X)$ if and only if $Y$ is so in $X$.
We show that if $Zsubseteq Ysubseteq X$, where $Y$ has property-$U$ ($SU$) in
$X$ then $Y/Z$ has property-$U$ ($SU$) in $X/Z$. On the other hand $Y$ has
property-$SU$ in $X$ if $Y/Z$ has property-$SU$ in $X/Z$ and $Z (subseteq Y)$
is an M-ideal in $X$. It is observed that a smooth Banach space of dimension
$>3$ is a Hilbert space if and only if for any two subspaces $Y, Z$ with
property-$SU$ in $X$, $Y+Z$ has property-$SU$ in $X$ whenever $Y+Z$ is closed.
We characterize all hyperplanes in $c_0$ which have property-$SU$. | | Source: | arXiv, 2009.09581 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser claudebot
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER