Understanding purpose of dual vector space

I am a beginner in differential geometry and have questions/confusion about dual vector space. I took a look at this and this questions. But both did not resolve my question.

We have standard definition of a dual vector space as:

Let $G,H$ are real vector spaces, we define vector space of all linear maps $f : G \rightarrow H$. Dual space $G^*$ is defined as $G^*: G \rightarrow \mathbb R$

• Sometimes notation $HOM(G,H)$ is used to indicate vector space of all linear maps. Does this have to do with homomorphism? Can someone comment what is role of homomorphism here?
• When $G,H$ are both real vector spaces, why can't we say dual space $G^*$ is $G^*: G \rightarrow H$?
• I did not understand the purpose of defining another vector space mapping from original vector space to space of real numbers. I can guess, one such purpose is for non Cartesian coordinate systems. We have a vector as $\overline{ a} = a_{i}e^i = a^je_j$. Here $e^i,e_j$ are dual basis vectors. Is this the purpose for dual vector space?

I appreciate any inputs and thanks in advance!

• One reason that we care about dual vector spaces is that we can identify $\operatorname{Hom}(G, H)$ with the tensor product $G^* \otimes H$. Relatedly (and using the fact that $H^{**}$ is canonically isomorphic to $H$ for finite-dimensional $H$), any linear map $f: G \to H$ can be canonically identified with a map $f^*: H^* \to G^*$, namely, $(f^*(\phi))(g) := \phi(f(g))$. – Travis Willse Dec 29 '15 at 2:05

Sometimes notation $HOM(G,H)$ is used to indicate vector space of all linear maps. Does this have to do with homomorphism? Can someone comment what is role of homomorphism here?

Definition: A linear homomorphism is a linear map between vector spaces.

Definition: $\operatorname{Hom}(V,W)$, where $V$ and $W$ are two $\Bbb R$-vector spaces, is the collection of all linear homomorphisms from $V$ to $W$. With the usual notions of addition and scalar multiplication of functions $\operatorname{Hom}(V,W)$ is itself an $\Bbb R$-vector space.

Definition: The dual space to an $\Bbb R$-vector space $V$ is defined as $V^* = \operatorname{Hom}(V,\Bbb R)$.

When $G,H$ are both real vector spaces, why can't we say dual space $G^*$ is $G^*: G \rightarrow H$?

I don't see how you think this should work. For instance, how would you choose the correct $H$? The actual definition of the dual space is given above.

I did not understand the purpose of defining another vector space mapping from original vector space to space of real numbers. I can guess, one such purpose is for non Cartesian coordinate systems. We have a vector as $\overline{ a} = a_{i}e^i = a^je_j$. Here $e^i,e_j$ are dual basis vectors. Is this the purpose for dual vector space?

One immediate application of the dual space is indeed that it provides a nice way to expand vectors in a non-orthogonal basis.

Let $V$ be an $n$-dimensional $\Bbb R$-vector space. If $\{e_1, \dots, e_n\}\subset V$ is some (not necessarily orthogonal) basis for $V$ and $\{f^1, \dots, f^n\}\subset V^*$ is its dual basis, then for any vector $v\in\Bbb V$ we have $$v = \langle v, f^1\rangle e_1 + \cdots + \langle v, f^n\rangle e_n$$ where $\langle v, f^i\rangle := f^i(v)$.

Be careful though. It's not true that $a_{i}e^i = a^je_j$ as the LHS and RHS are two different types of objects (existing in two different vector spaces). I've seen physicists state this before but strictly speaking $V$ and $V^*$ are always distinct spaces (I think).

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\eps}{\varepsilon}$I'll briefly address the third question (the purpose of considering the dual space), since that's the most nebulous, subtle, and important in differential geometry.

In the setting of vector calculus (on a smooth $n$-dimensional manifold $M$), you have notions of local coordinates, change of coordinates, smooth functions, and smooth curves.

A smooth curve $\gamma:(-\eps, \eps) \to M$ defines a tangent vector $v = \gamma'(0)$ at the point $p = \gamma(0)$. The set of all tangent vectors at $p$ turns out to be an $n$-dimensional vector space, the tangent space $T_{p}M$.

A smooth, real-valued function $f:M \to \Reals$ defines a differential $df$. The "value" $df(p):T_{p}M \to \Reals$, which satisfies $$\frac{d}{dt}\bigg|_{t = 0} (f \circ \gamma)(t) = \bigl[df(p)\bigr](v),$$ is linear in $v$, and therefore constitutes a covector, i.e., an element of the vector space dual to $T_{p}M$.

A finite-dimensional real vector space $V$ and its dual space $V^{*}$ have the same dimension, and so are isomorphic. However, there is no basis-independent way to define an isomorphism of $V$ with $V^{*}$. Consequently, vector-valued functions on $M$ (a.k.a. vector fields) and covector-valued functions on $M$ (a.k.a. differential one-forms) are distinct types of entities. For instance:

• If $\phi:M \to N$ is a smooth map of smooth manifolds, a one-form on $N$ pulls back (in a coordinate-invariant way) to a one-form on $M$, but a vector field on $N$ does not pull back in a coordinate-invariant way to a vector field on $M$. (A tangent vector on $M$ naturally pushes forward to a tangent vector on $N$, but a vector field on $M$ does not naturally push forward to a vector field on $N$ in general.)

• If $f:M \to \Reals$ is smooth, the differential $df$ depends only on the smooth structure of $M$, but there is no natural way to define a gradient vector field of $f$.

• A smooth curve in $M$ has a natural notion of velocity (first derivative), but has no natural notion of acceleration (second derivative).

This list is in no way comprehensive. All such items come down to the chain rule (or, in fancy language, to the transition functions for the tangent and cotangent bundles of $M$), see also Dual space and covectors: force, work and energy for detailed computations along similar lines.

The second and third items are often avoided by working on a Riemannian manifold, equipped with a Riemannian metric $g$ (which can be used to define an isomorphism between vectors and covectors) and its Levi-Civita connection (which defines covariant differentiation, such as acceleration of a curve).

The fact that a function has a gradient field and a smooth curve has an acceleration in elementary vector calculus secretly relies on the Euclidean metric on Cartesian space.

Tensor Analysis on Manifolds by Bishop and Goldberg, and Comprehensive Introduction to Differential Geometry, Volume I by Spivak, are excellent sources for further reading. (There are surely more recent treatments, as well.)