Exterior algebra of a Banach space
In this carefully written paper, based on results of the author's Ph.D.\ thesis, the exterior (Grassmann) algebra \(\Lambda_\ast X\) of a Banach space \(X\) is introduced and equipped with several norms, among them the projective and the injective norms \(\|\cdot \|_\wedge\) and \(\|\cdot \|_\vee\), respectively (which are related to the homonymous norms by Grothendieck). Several universal properties of these normed spaces are shown, and it turns out that \((\Lambda_\ast X)_{\wedge}\) is a Banach algebra. As explicit examples, the author shows that \((\Lambda_\ast \ell_1)_{\wedge}=\ell_1\) and \((\Lambda_\ast \ell_2)_{\wedge}\not=\ell_2\). It is also shown that the norms \(\|\cdot \|_\wedge\) and \(\|\cdot \|_\vee\) are in duality.
scientific article; zbMATH DE number 5149358