TITLE: What is the difference between vector space and dual space?
QUESTION [7 upvotes]: I read that in Dirac notation, kets are elements of a vector space and bras are elements of the dual space. My question is, what is the difference between vector space and dual space, and why are bras elements of the dual space?
Thank you.

REPLY [2 votes]: For the finite dimensional case we have for a column vector $u$:
$$
u = \vert u \rangle\\
u^+= \langle u \vert
$$
where $u^+$ is the transposed, complex conjugated, thus adjugated, vector.
It is a linear form from $V$ to the scalar field.
If $V$ is a vector space then $V^*$, consisting of all linear forms of $V$, is a linear vector space as well.