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.

 

[wrobel]

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 eq

Hope you are speaking about Hilbert spaces. If it is so you should separate two cases: a Hilbert space over $\mathbb{R}$ and over $\mathbb{C}$. In the second case be careful with duality. For details see for example Yosida: Functional Analysis, Riesz Representation Theorem and near it

 

[wrobel]

On the other hand consider a real Hilbert space $F$ of sequences $x=(x_1,x_2,\ldots)$ with inner product
$$(x,y)_F=\sum_{k=1}^\infty k^2x_ky_k,\quad \|x\|_F^2=\sum_{k=1}^\infty k^2x_k^2<\infty.$$
It is easy to see that the dual space $F'$ consists of sequences
$x'=(x_1',\ldots)$ such that
$$\|x'\|_{F'}^2=\sum_{k=1}^\infty\frac{1}{k^2}x_k'^2<\infty,\quad \langle x',y\rangle=\sum_{k=1}^\infty x'_ky_k,\quad y\in F.$$
The space $F'$ is a real Hilbert space:
$$(x',y')_{F'}=\sum_{k=1}^\infty\frac{1}{k^2}x_k'y'_k.$$

 

[nomadreid]

Thanks for the replies, Gavran and Wrobel.

First, Gavran. That is good to hear. Could you give me an example of each, and why? (That is, one in which it is better to use an inner product, on in which it is better to use a dual space operation.) That would be super.

Second, Wrobel. I am more or less familiar with Riesz's theorem, but it was my impression that it provided the correspondence between the two (inner product and dual space.) I will look at Yosida (on your suggestion I got the book), but could you tell me what I should look for that will tell me when one is better/necessary to use one (inner product, dual space) than the other? Further, yes, I was thinking of Hilbert spaces, so I am intrigued what you meant by being careful with duality with a complex field. Finally, your example seems to be providing an example of an inner product and its corresponding dual space; I am not sure how I would know when one of the two would be more advantageous to use than the other. Many thanks in advance for further clarification.

 

[wrobel]

I was thinking of Hilbert spaces, so I am intrigued what you meant by being careful with duality with a complex field.

See Chapter 3 Section 6: F. Riesz' Representation Theorem and Corollary 1 therein. In the case of $\mathbb{C}$ the isomorphism between $X$ and $X'$ is not linear but conjugate linear.

Finally, your example seems to be providing an example of an inner product and its corresponding dual space

Sometimes (in Sobolev spaces for example) the dual space can be realized in nontrivial way. In this example $F$ and $F'$ are very different spaces:
$$(1,1,\ldots)\in F',\quad (1,1,\ldots)\notin F.$$
But $F$ and $F'$ are isomorphic certainly

 

[Gavran]

First, Gavran. That is good to hear. Could you give me an example of each, and why? (That is, one in which it is better to use an inner product, on in which it is better to use a dual space operation.) That would be super.

Dealing with a dual vector space lacks bilinearity. It is more natural to use an inner product when we want to define lengths, distances, and angles; when we want to identify perpendicular vectors; and when we want to decompose vectors onto subspaces.
On the other hand, an inner product lacks linearity. It is more natural to use a dual vector space as a tool when we want to deal with linear equations or when we want to deal with operators (gradients, integrals…).
A dual vector space is a more general concept than an inner product because every vector space has a dual vector space, but every vector space does not have an inner product.

 

[Gavran]

Dual vector spaces make analyzing and understanding vector spaces with no inner products more clear.
Let us consider the set of real numbers R as a vector space over the field of rational numbers Q. The standard inner product does not exist, but we can introduce a dual space R* as a set of mappings from R to Q. R as a vector space over Q has a dimension of the cardinality of R, and its dual vector space has a dimension of the cardinality of the power set of R. So we can say that R over Q and its dual vector space are infinite-dimensional, and the dual space has a basis that is larger in its cardinality.

 

[nomadreid]

Huge thanks for the replies, Gavran. (The delay in my response was due to my not getting the usual email notification of a reply.) Your examples and explanations are excellent, and well answer my question. Super!

Also thanks to you, Wrobel, for the warning, and for your interesting example.