## View abstract

### Session S05 - Advances on Spaces of Non-absolutely Integrable Functions and Related Applications

Monday, July 12, 14:55 ~ 15:25 UTC-3

## An application of a theorem of Kuelbs' to a characterization of complemented subspaces of a separable Banach space

### Douglas Mupasiri

#### University of Northern Iowa, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak12ce3d912a5b0d29cc16eda7b33d5927').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy12ce3d912a5b0d29cc16eda7b33d5927 = 'd&#111;&#117;gl&#97;s.m&#117;p&#97;s&#105;r&#105;' + '&#64;'; addy12ce3d912a5b0d29cc16eda7b33d5927 = addy12ce3d912a5b0d29cc16eda7b33d5927 + '&#117;n&#105;' + '&#46;' + '&#101;d&#117;'; var addy_text12ce3d912a5b0d29cc16eda7b33d5927 = 'd&#111;&#117;gl&#97;s.m&#117;p&#97;s&#105;r&#105;' + '&#64;' + '&#117;n&#105;' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak12ce3d912a5b0d29cc16eda7b33d5927').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy12ce3d912a5b0d29cc16eda7b33d5927 + '\'>'+addy_text12ce3d912a5b0d29cc16eda7b33d5927+'<\/a>';

In this talk we apply a theorem of Kuelbs' on continuously and densely embedding separable Banach spaces in Hilbert spaces to obtain a necessary and sufficient condition for a closed, proper subspace of a separable Banach space to be complemented. To motivate our approach we first show that $c_0$ is dense in $\ell_\infty$ with respect to the topology of a Hilbert space, $\mathcal{H}$, containing $\ell_\infty$. We use this fact to give an alternative proof to the well-known result, due to R. S. Phillips (1940), that $c_0$ is not complemented in $\ell_{\infty}$. We apply the idea of our proof of this result to show that if $A$ is a separable Banach space and $B$ is a closed subspace of $A$, then $B$ is complemented in $A$ if and only if $B$ is not dense in $A$ with respect to the topology of the (separable) Kuelbs Hilbert space determined by $A$

Joint work with Tepper Gill (Howard University, USA).