2-Rotund norms for unconditional and symmetric sequence spaces

Area of Science:

• Functional Analysis
• Banach Space Theory

Background:

• Banach spaces are fundamental in functional analysis.
• Unconditional and symmetric bases are key structural properties of Banach spaces.
• Renorming techniques are crucial for exploring geometric properties of Banach spaces.

Purpose of the Study:

• To investigate the existence of equivalent 1-unconditional 2R norms in reflexive Banach spaces with unconditional bases.
• To explore embeddings of such spaces into reflexive spaces with 1-symmetric 2R norms.
• To derive partial results on 1-symmetric 2R renormings for spaces with symmetric bases.

Main Methods:

• Utilizing properties of reflexive Banach spaces.
• Applying concepts of unconditional and symmetric bases.
• Developing and applying 2R renorming techniques.

Main Results:

• Demonstrated that a reflexive Banach space with an unconditional basis admits an equivalent 1-unconditional 2R norm.
• Proved that such spaces embed into a reflexive space equipped with a 1-symmetric 2R norm.
• Obtained partial findings concerning 1-symmetric 2R renormings for spaces with symmetric bases.

Conclusions:

• The study establishes a connection between unconditional bases and the existence of 1-unconditional 2R norms.
• It provides insights into the renorming of Banach spaces and their embeddings.
• The results contribute to the understanding of geometric properties in Banach space theory.