Hello, The next Banach spaces webinar is on Friday November 5 9AM Central time. Please join us at https://unt.zoom.us/j/83807914306 Title: $L$-orthogonal elements and spaces of operators Speaker: Abraham Rueda Zoca (Universidad de Murcia) Abstract. Given a Banach space $X$, we say that an element $u\in X^{**}$ is $L$-orthogonal if, for every $x\in X$, it follows that $$\Vert x+u\Vert=\Vert x\Vert+\Vert u\Vert.$$ In 1989, G. Godefroy proved that a Banach space $X$ admits an equivalent renorming with non-zero $L$-orthogonal elements if, and only if, $X$ contains an isomorphic copy of $\ell_1$. Moreover, G. Godefroy and N. J. Kalton proved (in 1989 too) that a separable space $X$ has non-zero $L$-orthogonal elements if, and only if, the following condition holds: \begin{center} For every finite-dimensional subspace $F$ of $X$ and every $\varepsilon>0$ there exists $x\in S_X$ so that $\Vert y+\lambda x\Vert\geq (1-\varepsilon)(\Vert y\Vert+\vert\lambda\vert)$ holds for every $y\in F$ and every $\lambda\in\mathbb R$. \end{center} In this talk we will examine the validity of this theorem for non-separable Banach spaces. For this, and for other results of the structure of the set of $L$-orthogonal elements, the Banach spaces of linear bounded operators between two Banach spaces will play a crucial role. The author was supported by Juan de la Cierva-Formaci\'on fellowship FJC2019-039973, by MTM2017-86182-P (Government of Spain, AEI/FEDER, EU), by MICINN (Spain) Grant PGC2018-093794-B-I00 (MCIU, AEI, FEDER, UE), by Fundaci\'on S\'eneca, ACyT Regi\'on de Murcia grant 20797/PI/18, by Junta de Andaluc\'ia Grant A-FQM-484-UGR18 and by Junta de Andaluc\'ia Grant FQM-0185. For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach Thank you, and best regards, Bunyamin Sari