On Extensions of Bilinear Maps

TL;DR

This paper characterizes when bounded bilinear maps can be extended from subspaces to entire Banach or Hilbert spaces, with applications to projective tensor products, providing a comprehensive understanding of such extensions.

Contribution

It provides necessary and sufficient conditions for extending bounded bilinear maps on subspaces of Banach and Hilbert spaces, including a full characterization for Hilbert spaces.

Findings

Necessary and sufficient conditions for extensions of bilinear maps.

Full characterization for Hilbert space subspaces.

Applications to projective tensor products.

Abstract

The paper deals with extension of bounded bilinear maps$.$ It gives a necessary and sufficient condition for extending a bounded bilinear map on the Cartesian product of subspaces of Banach spaces$.$ This leads to a full characterization for extension of bounded bilinear maps on the Cartesian product of arbitrary subspaces of Hilbert spaces$.$ Applications concerning projective tensor products are also investigated.