My question is , What is the suitable example of a Banach space that In an infinite-dimensional Banach space, the kernel of any discontinuous linear functional is a barrelled subspace (this subspace is an incom plete barrelled normed space).
2026-04-04 11:07:26.1775300846
About barrelled subspace.
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According to proposition 4.3.1 in the book Barrelled Locally Convex Spaces of Bonet and Perez Carreras every finite-codimensional subspace (in particular, every kernel of a linear functional) of a barrelled space is again barrelled.
You can thus take any infinite dimensional Banach space and any discontinuous linear functional (which exists if you believe in the axiom of choice) to construct incomplete barrelled spaces.