Realisationstheoreme

The Realisation Theorem, also known as the Realisation Theorem for Banach Spaces, is a fundamental result in functional analysis, a branch of mathematics that studies vector spaces equipped with a norm. The theorem provides a characterisation of Banach spaces in terms of their dual spaces and their preduals. A predual of a Banach space is a Banach space whose dual is isomorphic to the original space.

The theorem states that a Banach space is isometrically isomorphic to the dual of its predual. This means that for any Banach space, there exists another Banach space (its predual) such that the original space is isometrically isomorphic to the dual of this predual space. The isomorphism is isometric, meaning it preserves the norm of the vectors.

The Realisation Theorem is significant because it provides a way to understand the structure of Banach spaces through their duals and preduals. It is a cornerstone in the study of Banach spaces and has applications in various areas of mathematics, including operator theory, harmonic analysis, and the theory of Banach algebras.

The theorem was first proven by Banach himself in the 1930s, and it has since been generalised and extended in various ways. It is a testament to the power and elegance of functional analysis, providing deep insights into the nature of infinite-dimensional spaces.