[Math] the difference between vector space and dual space

Question

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.

Answer 1

A vector space over a field $\mathbb{F}$ is a set $V$ with operations $+$ and $\cdot$ satisfying the vector space axioms. Given a vector space $V$, it's dual space $V^\star$ is defined as $\mathrm{Hom}(V,\mathbb{F})$, i.e. the set of all linear maps(functionals) between the vector space and its underlying field(considered as an own vector space in this case).

Normally, the Dirac notation $\langle v|w\rangle$ is a representation of a scalar product and the Bra and Ket correspond on a low level to vectors input in this scalar product. Note, that for the complex scalar product, order of the arguments is important due to its hermitian nature and its semi-bilinearity, i.e. that it is conjugate linear w.r.t. to one argument. A classical way is to define the standard complex scalar product to be conjugate linear in the second argument. In Bra-Ket-notation, you usually reverse the order of the arguments of the complex scalar product, i.e. $\langle x,y\rangle=\langle y|x\rangle$ resulting in conjugate linearity in the first argument.

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The key thing is that there is a strong correspondence between scalar products and members of the dual space, i.e. linear functionals.

There is a way that any functional corresponds in a one-to-one fashion to a representation using the scalar product(of the associated vector space). This is known as Riesz representation theorem.

More precisely, you may look at a vector $v\in V$ and suppose that $\langle\cdot,\cdot\rangle$ is an associated scalar product(turning $V$ into a euclidean/unitary space in the real/complex case). Then the map $\varphi_v:w\mapsto\langle w,v\rangle$ is a member of the dual space $V^\star$ and the theorem says that any linear functional $\psi\in V^\star$ can be written as such a $\varphi_v$ uniquely.

Thus, you may convert a scalar product between two vectors into an application of a linear functional to another vector, pulling the problem statement into the realm of dual spaces, where you have other mathematical possibilities to tackle various questions.

EDIT: Note, that the definition of $\varphi_v$ of course depends also on the argument which is assumed to be conjugate linear, in this case the second, as linearity in the first is needed to make $\varphi_v$ a linear map(check this).