Which definition of dual space is more common in research?

Question

Generally, whenever I ask a question or talk about dual space of a vector space, people tend to understand this space as the space of linear functions from the vector space to the field. Lots of examples of this situation even in this site.The reason for this obviously this term commonly used for this space in at least undergraduate courses.

However, the book such as Linear algebra by Werner Greub, which is a graduate text, defined dual space of a vector space as the space for which a non-degenerate bilinear function is defined between $E$ and $E^*$ (dual space), which is more general than the first definition that I have mentioned.

Hence, my question is that between the researchers (mathematicians, physicist etc.) how is this term used ? I mean in which sense that is used ? In other words, if I see the term dual space in a research paper, which space should I consider ?

Answer 1

I have never seen anyone use that second definition. Among other things, it is not functorial. (However, it is related to a more general notion of dual object in category theory.)

Answer 2

According to the Wikipedia article List of dualities, there are 53 (!) different kind of dualities used in mathematics. Therefore, depending on the topic you are working on, the term dual space might have different meanings.

That being said, if you restrict to vector spaces, there seems to be only two accepted possibilities, the algebraic dual space and the continuous dual space.