2 interpretations of bra-ket expression: equal, & isomorphic, but...

[nomadreid]

TL;DR

The inner product inside a vector space is equal to the dual of one being applied as a functional to the other. The first (inner product) involves a single vector space, and the second (functional) involves two. Hence the two are equal and isomorphic but not identical. Is there any time in which it is worthwhile to separate the two?

Starting with a vector space V equipped with an inner product (. , .), and its dual space V*, one can look at the expression <a|b> in one of two ways

It is the dot product ( |a> ,|b> ), with |a> and |b> from V

It is the functional <a| from V* applied to |b> from V.

Since the two equal the same scalar in the field, and because there is an isomorphism between the two, then it appears in practice that one goes back and forth between the two, and it doesn't seem to make any difference which way one thinks of it. Or is there? That is, are there cases in which a very pedantic mathematician might look at it one way rather than another? Alternatively: are there cases in which there is a good reason to think of it in one way rather than the other?

 

[Hill]

They are different, e.g., see https://www.clrn.org/is-inner-product-same-as-dot-product/.

The dot product is a specific instance of the inner product, applicable only in Euclidean spaces with the standard Euclidean metric, whereas inner products can be customized to fit different vector spaces and application requirements.

 

[nomadreid]

Thanks for answering, Hill, although it appears that either I did not state the question clearly enough, or you read the question rather hastily, as I mentioned neither which inner product was being referred to nor over what field the vector fields were; I neither mentioned nor meant the dot product. So I am afraid your reply did not address my question.

 

[Gavran]

Since the two equal the same scalar in the field, and because there is an isomorphism between the two, then it appears in practice that one goes back and forth between the two, and it doesn't seem to make any difference which way one thinks of it. Or is there? That is, are there cases in which a very pedantic mathematician might look at it one way rather than another? Alternatively: are there cases in which there is a good reason to think of it in one way rather than the other?

There is a difference.
The difference lies not only in the fact that an inner product is bilinear and a dual space operation is linear, but also in the purpose. Sometimes it is better to use an inner product, and sometimes it is better to use a dual space operation.