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An abstract algebraic-topological approach to the notions of a first and a second dual space. I. (English) Zbl 1235.46007
Di Maio, Giuseppe (ed.) et al., Theory and applications of proximity, nearness and uniformity. Caserta: Dipartimento di Matematica, Seconda Università di Napoli; Rome: Aracne (ISBN 978-88-548-2857-5/hbk). Quaderni di Matematica 22, 275-297 (2009).
Let $$(X,\|\cdot\|)$$ be a normed $$\mathbb K$$-vector space, where $$\mathbb K$$ is either the scalar field of reals $$\mathbb R$$ or the scalar field of complex numbers $$\mathbb C$$. In functional analysis, it is a standard fact that the first dual $$X^\prime = \{f: X \to \mathbb K: f$$ is linear and continuous$$\}$$ is a normed space, with the usual operations of sum and scalar multiplication, and the operator norm. Furthermore, the second dual is just defined as $$X''= (X^\prime)^\prime$$, and the normed space $$X$$ can be embedded into its second dual via the classical canonical duality mapping $$J: X \to X''$$ defined by $$J(x)(f)=f(x)$$ for all $$x\in X$$ and $$f \in X'$$. However, when $$X$$ has some algebraic structure, this approach to construct duals may fail. For example, when $$X$$ and $$Y$$ are rings, one thinks that the set $$X^d= \{h \in Y^X: h$$ is a ring homomorphism$$\}$$ defines the first dual of $$X$$ with respect to $$Y$$, with the usual pointwise operations. But, in general, $$X^d$$ is not closed under pointwise operations.
In this paper, the authors propose an abstract scheme to construct a first and a second dual space of a suitable space $$X$$. The advantage of this scheme is that it not only includes the classical approach in functional analysis for the construction of duals as a special case, but can also apply to those spaces of $$X$$ with certain algebraic structures.
For the entire collection see [Zbl 1202.00092].

##### MSC:
 46A20 Duality theory for topological vector spaces 46B10 Duality and reflexivity in normed linear and Banach spaces 46E25 Rings and algebras of continuous, differentiable or analytic functions 46J25 Representations of commutative topological algebras 54C35 Function spaces in general topology