Abstract
This paper deals with fuzzy normed spaces in the sense of Cheng and Mordeson. We characterize fuzzy norms in terms of ascending and separating families of seminorms and prove an extension theorem for continuous linear functionals on a fuzzy normed space. Our result generalizes the classical Hahn-Banach extension theorem for normed spaces.
Highlights
The Hahn-Banach extension theorem is without doubt one of the most important theorems in the whole theory of normed spaces
By omitting the condition of left continuity of the real function N(x, ⋅) in the definition of fuzzy norm given by Cheng and Morderson [5], Bag and Samanta gave in [7] the following notion
Bag and Samanta proved in Theorem 2.1 of [7] that if (X, N, ∧) is a BS-fuzzy normed space which satisfies condition (N6), {‖ ⋅ ‖α : α ∈ (0, 1)} is an ascending family of norms on X, where
Summary
The Hahn-Banach extension theorem is without doubt one of the most important theorems in the whole theory of normed spaces. Bag and Samanta considered in [7] a fuzzy norm slightly different from this one and they proved a series of results that have been used in many subsequent works in this context. This definition is less restrictive than the one given by Cheng and Mordeson, the more interesting results given in the mentioned paper require the use of two very restrictive additional conditions which leave out of the scope of these results important examples of fuzzy normed spaces.
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