Intuitions on a Dual Vector Space

[sineontheline]

So I'm pretty sure I understand the formalism of dual vector spaces. (E.g. there exist objects that operate on vectors and take them to scalars. these objects themselves form a linear vector space).

But I'm having difficulty understanding where this comes from intuitively. How would I know that I need them if I'm trying to reason things out?

More clearly:

1) I have a linear vector space (LVS).
2) I say "I have addition of vectors from condition of LVS. From my experience with the outside world, I know I need my space of vectors to have notions of length and distance, not just rules for combining them."
3) Ok, then I need a way to combine vectors in my space that have something to do with how they're positionally related to one another.
4) ?

Now what?
1) How do I know that the distance operation I'm looking for is the scalar product?
2) Why do I need a notion of distance to specify a scalar product? (v^uwⁿg_un => in order to have my scalar product to work, I need g_un)
3) Why do I not need a notion of distance if I use objects from the dual space?
(wⁿg_un = w_u => v^uw_u = scalar)

This might need to be in the GR forum, but I was reading the first chapter of Shankar for the billionth time when I was able to finally articulate all this. as you might be able to tell -- this is *really* bothering me, I'm really confused about the motivation for defining all this stuff. please hellllllp! <3

 

[strangerep]

1) How do I know that the distance operation I'm looking for is the scalar product?

Maybe think in terms of angles between rays?

2) Why do I need a notion of distance to specify a scalar product? (v^uwⁿg_un => in order to have my scalar product to work, I need g_un)
3) Why do I not need a notion of distance if I use objects from the dual space?
(wⁿg_un = w_u => v^uw_u = scalar)

The (nondegenerate) metric business only works because the spaces are isomorphic.
In the inf-dim case, the primal and dual spaces are not isomorphic in general so you
can't use a simplistic metric to convert between the two.

 

[sheaf]

I always liked the description of dual vectors which is given in Misner Thorne and Wheeler's "Gravitation". The idea is that a dual vector in an N dimensional vector space is a bunch of (N-1) dimensional parallel hyperplanes. The scalar you get from hitting a vector with a dual vector is a measure of the "number" of hyperplanes the vector pierces. No need for a metric to obtain the scalar.

Where you do need a metric, though, is if you want to map a vector to a dual vector or vice versa.

But maybe you knew all this and it was something different you were asking...

 

[sineontheline]

The (nondegenerate) metric business only works because the spaces are isomorphic.
In the inf-dim case, the primal and dual spaces are not isomorphic in general so you
can't use a simplistic metric to convert between the two.

This is interesting. So its not always true that V** = V (dual of the dual is the vector space I start with)? What's the intuition for why it's true for certain cases but not for others? What makes Hilbert Spaces so special? Do Fourier spaces have this property?

Where can I find more information about this?

 

[bapowell]

I prefer to think of the dual space simply as the space of linear functionals that act on the tangent space, i.e. they are maps that send vectors to the reals,

$$\omega: TM \rightarrow R$$

where here $\omega \in T^{*}M$ is a one-form in the cotangent (dual) space, $TM$ is the tangent space, and $R$ is the reals. From this definition, the space of 0-forms is just the space of scalar functions on the reals.

This is the most natural way to construct an inner product on a manifold. A great reference on differential forms (dual vectors) is the Dover book by Flanders. Forms as used in GR are covered fairly well in a number of places, like Nakahara's book Geometry, Topology, and Physics.

 

[strangerep]

This is interesting. So its not always true that V** = V (dual of the dual is the vector space I start with)?

If the primal space V is isomorphic to its bidual V**, then V is called "reflexive".
All the interesting cases of interest in physics that I know of are indeed reflexive.
I've been told that examples of nonreflexive spaces are nonconstructive,
but I don't know any more about them than that, sorry.

What's the intuition for why it's true for certain cases but not for others?
What makes Hilbert Spaces so special?
Do Fourier spaces have this property?
Where can I find more information about this?

I'm not sure what you mean by "Fourier spaces". The Fourier transform is
simply a change of basis in an infinite-dimensional space.

Try Ballentine ch1, esp. section 1.4 for a gentle introduction to the basics
of dual spaces. Beyond that, you'll need a book on functional analysis and/or
generalized functions (distributions) -- but they tend to be heavy going.

 

[Talisman]

This is interesting. So its not always true that V** = V (dual of the dual is the vector space I start with)? What's the intuition for why it's true for certain cases but not for others? What makes Hilbert Spaces so special? Do Fourier spaces have this property?

Where can I find more information about this?

For any finite dimensional V, it's true that V is (canonically) isomorphic to V**; the isomorphism is given by v**(v*) = v*(v).

For an intuitive explanation of why it fails in the infinite-dimensional case, I highly suggest this page by Tim Gowers:

http://www.dpmms.cam.ac.uk/~wtg10/meta.doubledual.html

In particular, read the section titled "Infinite-dimensional vector spaces."