# Dual and Generalized Dual Cones in Banach Spaces

## TL;DR

This paper explores dual cones, faces, and projections in Banach spaces, highlighting differences from Hilbert spaces and providing illustrative examples to deepen understanding.

## Contribution

It introduces and analyzes dual cones, faces, and visions in Banach spaces, extending concepts from Hilbert space theory and examining their properties.

## Key findings

- Dual cones lose key properties when moving from Hilbert to Banach spaces
- Relations between faces, visions, and projections are established in Banach spaces
- Illustrative examples demonstrate the theoretical results

## Abstract

The primary objective of this paper is to propose and analyze the notion of dual cones associated with the metric projection and generalized projection in Banach spaces. We show that the dual cones, related to the metric projection and generalized metric projection, lose many important properties in transitioning from Hilbert spaces to Banach spaces. We also propose and analyze the notions of faces and visions in Banach spaces and relate them to the metric projection and generalized projection. We provide many illustrative examples to give insight into the given results.