Bipolar Theorem and Some of Its Applications in Fuzzy Quasi‐Normed Space

The classical bipolar theorem plays an important role in functional analysis. This paper generalizes this theorem to fuzzy quasi‐normed spaces, which include asymmetric normed space and fuzzy normed space as special cases. First, the concept of the asymmetric polar of a subset is introduced in the fuzzy quasi‐normed space, its basic properties, such as closedness and compactness, are investigated. After the notion of asymmetric bipolar being proposed, the bipolar theorem is established. Additionally, some conclusions are presented based on the bipolar theorem. For example, a necessary and sufficient condition for the linear hull of a subset to be dense is given, a representation of the gauge of a subset is presented, and a characteristic of the family of equicontinuous linear functional is proved. These results generalize the existing results.