# Smoothness and norm attainment of bounded bilinear operators between Banach spaces

## TL;DR

This paper investigates the smoothness and norm attainment of bounded bilinear operators between Banach spaces, providing characterizations using Birkhoff-James orthogonality and semi-inner-products, especially in finite-dimensional and smooth cases.

## Contribution

It offers new characterizations of smoothness and norm attainment for bilinear operators using orthogonality and semi-inner-products, extending to infinite-dimensional spaces.

## Key findings

- Characterization of Birkhoff-James orthogonality in finite dimensions
- Complete description of norm attainment set using semi-inner-products
- Smoothness criteria for bilinear operators in Banach spaces

## Abstract

We study the smoothness and the norm attainment of bounded bilinear operators between Banach spaces, using the concepts of Birkhoff-James orthogonality and semi-inner-products. In the finite-dimensional case, we characterize Birkhoff-James orthogonality of bilinear operators in terms of the norm attainment set. This yields a nice characterization of smoothness of bilinear operators between Banach spaces, in the finite-dimensional case. Without any restriction on the dimension of the space, we obtain a complete characterization of the norm attainment set of a bounded bilinear operator using semi-inner-products, which is particularly useful when the concerned Banach spaces are smooth.