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 \({v=v^{\mu}e_{\mu}\mapsto (\eta_{\mu\lambda}v^{\mu})\beta^{\lambda}\equiv v_{\lambda}\beta^{\lambda}}\), which is identical 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.

Now, 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 \({\varphi^{\sharp}}\). If we then define \({\varphi^{\Uparrow}\equiv\varphi^{\sharp}/\left\Vert \varphi^{\sharp}\right\Vert ^{2}}\), the length of this projection as a multiple of \({\left\Vert \varphi^{\Uparrow}\right\Vert}\) is \({\left\langle \varphi^{\Uparrow},v\right\rangle /\left\Vert \varphi^{\Uparrow}\right\Vert =\varphi(v)}\). We can therefore represent a 1-form \({\varphi}\) as a “receptacle” \({\varphi^{\Uparrow}}\) which when applied to a vector “arrow” argument \({v}\) yields the number of receptacles covered by the projection of \({v}\) onto \({\varphi^{\sharp}}\), which is the value of \({\varphi(v)}\). The advantage of this approach is that values can be calculated from a figure absent a length scale. Another common graphical device is to represent \({\varphi}\) as a density of “surfaces” where the value of \({\varphi(v)}\) is the number of surfaces “pierced” by the arrow. The following figure covers some non-intuitive aspects of these visualizations.

Depicting a 1-form \({\varphi}\) as the associated vector \({\varphi^{\Uparrow}}\) or as a density of surfaces has consequences that can be non-intuitive. When orthogonality corresponds to right angles in a figure, an orthonormal basis and its dual basis appear as identical arrows; in the figure, we see that for a non-orthonormal basis, the dual basis does not appear to either be parallel to the basis or to have identical lengths. We also see that quadrupling the value of the 1-form means quartering its length in the figure, or equivalently quadrupling the density of surfaces pierced by arrows. This means that when depicting a linearly changing 1-form as above, the length \({L}\) of the associated vector changes like \({L\mapsto L/(1+r\varepsilon)}\) for some scaling factor \({r}\), which doesn’t appear linear as a vector representation would, whose length changes like \({L\mapsto L(1+r\varepsilon)}\).

Δ It is important to remember that the practice of depicting a 1-form \({\varphi}\) as the associated vector \({\varphi^{\Uparrow}}\) or as a density of surfaces has consequences that can be non-intuitive.

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