Differing definitions of an inner product

In summary, there are two main definitions of an inner product on a vector space - one as a bilinear map on the vector space itself, and one as a bilinear map on the dual space of the vector space. While the first definition is commonly used for Riemannian metrics, the second definition is more general and is used in the formalism of the Riesz Representation theorem. The difference in notation is due to the isomorphism between the vector space and its dual, and the second definition does not induce any extra structure on the vector space. However, it is important to note that the notation used for the natural product on the dual space and vector space may sometimes be interchangeable, but it is not entirely accurate to call the
  • #1
Kreizhn
743
1
Hey all,

This might seem like a stupid question, and this might not be the correct forum, but hopefully someone can clarify it really easily.

I often have seen two definitions of an inner product on a vector space. Firstly, it can be defined as a bilinear map on a [itex] \mathbb F-[/itex]vector space V as
[tex] \langle \cdot, \cdot \rangle : V \times V \to \mathbb F[/itex]
satisfying the usual inner product conditions. An example that comes to mind is the Riemannian metric, which is a 2-tensor and so acts on two copies of a tangent space. Alternatively, I've seen it defined as
[tex] \langle \cdot, \cdot \rangle : V^* \times V \to \mathbb F [/tex]
satisfying the usual inner product conditions. An example that comes to mind here is the formalism used in the Riesz Representation theorem.

The only place I've really seen the first definition is in the case of Riemannian metrics, hence the motivation for posting this discussion in this forum.

Now I know that for finite dimensional vector spaces that V and [itex] V^* [/itex] are isomorphic: is this the reason for the differing notations? Or is it perhaps that in the second case when the domain is [itex] V^*\times V[/itex] we've some how canonically identified a vector [itex] v \in V [/itex] with its induced linear functional
[tex] v \mapsto \langle v, \cdot \rangle [/tex]?

Again, this may seem really simple but I'd appreciate any response.
 
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  • #2
The pairing

[tex]
\langle \cdot, \cdot \rangle : V^* \times V \to \mathbb F
[/tex]

i s natural and it is not inducing an extra structure on V. On the other hand

[tex]
\langle \cdot, \cdot \rangle : V \times V \to \mathbb F
[/tex]

is essentially the same as fixing a particular identification of V and V* (though it is somewhat more complicated for sesquilinear scalar products).
 
Last edited:
  • #3
Are those supposed to be the same domains? And is there a particular identification, like the one I mentioned in my original post?
 
  • #4
I have made an error in my comment (unnecessary star in the second formula). Now corrected.
 
  • #5
Kreizhn said:
Alternatively, I've seen it defined as
[tex] \langle \cdot, \cdot \rangle : V^* \times V \to \mathbb F [/tex]
satisfying the usual inner product conditions. An example that comes to mind here is the formalism used in the Riesz Representation theorem.
While I have seen the natural product on V* x V written with that notation,
I have never seen it called an "inner product", except when being sloppy. The big point of the Riesz Representation theorem is that for the particular topological inner product space of interest, every continuous linear functional is the transpose of a vector.
 

1. What is the definition of an inner product?

The definition of an inner product is a mathematical operation that takes two vectors as inputs and produces a scalar value as an output. It is used to measure the angle between two vectors and the length of a vector. The most common definition of an inner product is the dot product, but there are other definitions as well.

2. How does the definition of an inner product differ from the dot product?

The dot product is the most commonly used definition of an inner product, but it is not the only one. The dot product only works for vectors in Euclidean space, while other definitions of an inner product can be used for more general vector spaces.

3. What are some examples of other definitions of an inner product?

Some examples of other definitions of an inner product include the Hermitian inner product, the sesquilinear inner product, and the bilinear inner product. These definitions are used for vector spaces such as complex vector spaces and function spaces.

4. How do differing definitions of an inner product impact calculations?

The choice of inner product definition can impact calculations in terms of the resulting values and interpretations. For example, the dot product will always result in a positive scalar value, while other definitions may result in negative or complex values. This can also affect the geometric interpretation of the angle between vectors.

5. How do differing definitions of an inner product impact applications in science?

The choice of inner product definition can impact applications in science, especially in fields such as physics and engineering where vector calculations are frequently used. Different definitions may be more appropriate for different types of vector spaces and can affect the accuracy and validity of calculations and interpretations.

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