The structure of the dual space

An element \({\varphi \colon V\to\mathbb{R}}\) of \({V^{*}}\) is called a 1-form. Given a pseudo inner product on \({V}\), we can construct an isomorphism between \({V}\) and \({V^{*}}\) defined by \({v\mapsto\left\langle v,\;\right\rangle }\), i.e. \({v\in V}\) is mapped to the element of \({V^{*}}\) which maps any vector \({w\in V}\) to \({\left\langle v,w\right\rangle}\). This isomorphism then induces a corresponding pseudo inner product on \({V^{*}}\) defined by \({\left\langle \left\langle v,\;\right\rangle ,\left\langle w,\;\right\rangle \right\rangle \equiv\left\langle v,w\right\rangle }\).

An equivalent way to set up this isomorphism is to choose a basis \({e_{\mu}}\) of \({V}\), and then form the dual basis \({\beta^{\lambda}}\) of \({V^{*}}\), defined to satisfy \({\beta^{\lambda}(e_{\mu})=\delta^{\lambda}{}_{\mu}}\). The isomorphism between \({V}\) and \({V^{*}}\) is then defined by the correspondence \({v=v^{\mu}e_{\mu}\mapsto (\eta_{\mu\lambda}v^{\mu})\beta^{\lambda}\equiv v_{\lambda}\beta^{\lambda}}\), corresponding to the isomorphism induced by the pseudo inner product on \({V}\) that makes \({e_{\mu}}\) orthonormal. Note that if \({\left\langle e_{\mu},e_{\mu}\right\rangle =-1}\) then \({e_{\mu}\mapsto-\beta^{\mu}}\). This isomorphism and its inverse (usually in the context of Riemannian manifolds) are called the musical isomorphisms, where if \({v=v^{\mu}e_{\mu}}\) and \({\varphi=\varphi_{\mu}\beta^{\mu}}\) we write

\begin{aligned} v^{\flat} & \equiv\left\langle v,\;\right\rangle \\
& =\left(\eta_{\mu\lambda}v^{\lambda}\right)\beta^{\mu}\\
& \equiv v_{\mu}\beta^{\mu}\\
\varphi^{\sharp} & \equiv\left\langle \varphi,\;\right\rangle \\
& =\left(\eta^{\mu\lambda}\varphi_{\lambda}\right)e_{\mu}\\
& \equiv\varphi^{\mu}e_{\mu}

and call the \({v^{\flat}}\) the flat of the vector \({v}\) and \({\varphi^{\sharp}}\) the sharp of the 1-form \({\varphi}\).

Δ It is important to remember that when the inner product is not positive definite, the signs of components may change under these isomorphisms. If the components are in terms of an arbitrary (non-orthonormal) basis, then as we will see, the components change their values as well, since \({\eta_{\lambda\mu}}\) is replaced by the metric tensor in the above analysis.

Note that since \({\beta^{\lambda}(e_{\mu})=\delta^{\lambda}{}_{\mu}}\) and \({\left\langle e_{\mu},e_{\lambda}\right\rangle =\eta_{\mu\lambda}}\) we have

\begin{aligned}\varphi(v)&=\varphi_{\lambda}\beta^{\lambda}(v^{\mu}e_{\mu})\\&=\varphi_{\mu}v^{\mu}\\&=\eta_{\mu\lambda}\varphi^{\lambda}v^{\mu} \\&=\left\langle \varphi^{\sharp},v\right\rangle .\end{aligned}

A 1-form acting on a vector can thus be viewed as yielding a projection. Specifically, with a positive definite inner product, \({\varphi(v)/\left\Vert \varphi^{\sharp}\right\Vert}\) is the length of the projection of \({v}\) onto the ray defined by \({\varphi^{\sharp}}\).

It is important to note that there is no canonical isomorphism between \({V}\) and \({V^{*}}\), i.e. we cannot uniquely associate a 1-form with a given vector without introducing extra structure, namely an inner product or a preferred basis. Either structure will do: a choice of basis is equivalent to the definition of the unique inner product on \({V}\) that makes this basis orthonormal, which then induces the same isomorphism as that induced by the dual basis.

In contrast, a canonical isomorphism \({V\cong V^{**}}\) can be made via the association \({v\in V\leftrightarrow\xi\in V^{**}}\) with \({\mathbb{\xi}\colon V^{*}\to\mathbb{R}}\) defined by \({\xi\left(\varphi\right)\equiv\varphi\left(v\right)}\). Thus \({V}\) and \({V^{**}}\) can be completely identified (for a finite-dimensional vector space), and we can view \({V}\) as the dual of \({V^{*}}\), with vectors regarded as linear mappings on 1-forms.

Vector components are often viewed as a column vector, which means that 1-forms act on vector components as row vectors (which then are acted on by matrices from the right). Under a change of basis we then have the following relationships:

Index notationMatrix notation
Dual basis\({\beta^{\prime\mu}=(A^{-1})^{\mu}{}_{\lambda}\beta^{\lambda}}\)\({\beta^{\prime}=A^{-1}\beta}\)
Vector components\({\left(v^{\mu}\right)^{\prime}=(A^{-1})^{\mu}{}_{\lambda}v^{\lambda}}\)\({v^{\prime}=A^{-1}v}\)
1-form components\({\left(\varphi_{\mu}\right)^{\prime}=A^{\lambda}{}_{\mu}\varphi_{\lambda}}\)\({\varphi^{\prime}=\varphi A}\)

Notes: We notationally distinguish between a changed vector \({e_{\mu}^{\prime}}\) and an unchanged vector with changed components \({\left(v^{\mu}\right)^{\prime}}\). A 1-form will sometimes be viewed as a column vector, i.e. as the transpose of the row vector (which is the sharp of the 1-form under a Riemannian signature). Then we have \({(\varphi^{\prime})^{T}=(\varphi A)^{T}=A^{T}\varphi^{T}}\).

An Illustrated Handbook