OK, walking the dogs last night I think I saw the problem. I warn you, this will be a lengthy post......

In what follows I use the so-called "summation convention" throughout - just as you did.

By definition, an element of an arbitrary vector space

is written as

where the

are scalars and the

are basis vectors.

Likewise an element in the dual to this space,

is written as

with the same general meaning.

Now if

is an inner product space, we have that

such that for any

there is some

where

, which is quite simply a non-negative number. One calls

a metric for this inner product space - it is a type (0,2) tensor.

Recall from the "raising and lowering indices" thread that

Now suppose

be a differential manifold, with

local coordinates for some point

I may define a vector space all of whose elements are tangent to

which one denotes by

. Now the basis vectors for this space are the operators

so by analogy with the above I seek some scalar multipliers.

Now since our manifold is differentiable, then

will do as well as another other, provided ONLY that

is a scalar valued function, since then

is a "genuine" scalar. So avector in

may be written as

(Remember I am assuming summation over like indices)

So, again by analogy, at each point

I may have a dual space called

whose basis vectors are

. Once again I choose the scalar multipliers to be

, so a covector, or one-form in

is just

Since we may not assume that our manifold is globally "flat" then the metric tensor may well be different at each point

, so let's write it in component form as, say

to acknowledge this fact

Putting this all together (if anyone out there hasn't lost the will to live) we will have that

Which is not quite exactly as Markus wrote - but I concede it was a valiant attempt