Banach Spaces with respect to Operator-Valued Norms

TL;DR

This paper introduces the concept of Banach and Hilbert spaces defined via operator-valued norms and inner products, expanding the framework of functional analysis with new examples and foundational notions.

Contribution

It defines L(H)-valued norms and inner products, establishing a new class of Banach and Hilbert spaces with several fundamental examples.

Findings

Introduction of L(H)-valued norms and Banach spaces

Definition of Hilbert spaces with L(H)-valued inner products

Provision of fundamental examples of these spaces

Abstract

We introduce the notions of L(H)-valued norms and Banach spaces with respect to L(H)-valued norms. In particular, we introduce Hilbert spaces with respect to L(H)-valued inner products. In addition, we provide several fundamental examples of Hilbert spaces with respect to L(H)-valued inner products.