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## Main Question or Discussion Point

So, I understand that the

Is it just that the two vector spaces form a

*dual space*, [itex]V^{\ast}[/itex] of a vector space [itex]V[/itex] over a scalar field [itex]\mathbb{F}[/itex] is the set of all linear functionals [itex]f^{\ast}:V\rightarrow\mathbb{F}[/itex] that map the vectors in [itex]V[/itex] to the scalar field [itex]\mathbb{F}[/itex], but I'm confused as to the meaning of the term*dual*?!Is it just that the two vector spaces form a

*dual pair*, such that there is a pairing between them [itex]\langle\cdot , \cdot\rangle : V^{\ast}\times V\rightarrow\mathbb{F}[/itex], that constructs a unique relation between them (i.e. each vector space has only one dual space), or is there some other reasoning to it?