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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 ?

Our
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  • It depends on the context. If you are a functional analyst,then you would probably consider "continuous" dual spaces. If you are an algebraist, you would probably consider "algebraic" dual spaces. – Batominovski Oct 01 '17 at 08:03
  • @Batominovski What do you exactly mean by that ? Can you elaborate a bit. – Our Oct 01 '17 at 08:04
  • The algebraic dual space of a vector space $V$ is the full dual: $V^*=\text{Hom}_{\mathbb{K}}(V,\mathbb{K})$. The continuous dual space depends highly on the topology. For example, the continuous dual space of the Banach space $L^p$ is $L^q$,where $p,q>0$ satisfy $\frac1p+\frac1q=1$. – Batominovski Oct 01 '17 at 08:07
  • @Batominovski But still it is considered as the space of functions with the domain as the vector space satisfying some properties, am I right ? – Our Oct 01 '17 at 08:08
  • I don't see a difference between the two definitions you are giving. Clearly, any element $e^$ of any space $E^$ that comes with a non-degenerate b.f. can be interpreted as a linear form on $E$ via $e\mapsto Q(\cdot, e^*)$ (and vice versa). – Peter Franek Oct 02 '17 at 09:07
  • @PeterFranek: You've written the induced a linear map from a Greub dual space to the commonly defined dual space, but this map might not be an isomorphism. –  Oct 02 '17 at 09:13
  • @Hurkyl Ok, I guess that it works in finite dimension, but in general I didn't know it is not the same thing. Thanks for your remark. – Peter Franek Oct 02 '17 at 16:29
  • @Our I'm probably violating some rule by commenting on this old question, but you might also be interested in my question here. – blargoner Jun 14 '21 at 03:18
  • @blargoner thanks for the link – Our Jun 14 '21 at 06:25

2 Answers2

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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.)

Arnaud D.
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Qiaochu Yuan
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  • It would be helpful if you also include which specific field are you working on. – Our Oct 01 '17 at 09:49
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    @onurcanbektas: look, it's not as if I'm totally unaware of what mathematicians in other fields do and say. I am telling you right now that I have never seen anyone, in any field, use your second definition, and I will bet, say, $100 that if you poll 10 graduate students or professors in any department in the United States, a majority of them will agree with me. Sometimes textbook authors just use idiosyncratic definitions; it's a thing that happens. – Qiaochu Yuan Oct 02 '17 at 06:52
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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.

J.-E. Pin
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