Natural isomorphism from V to V

In summary, there is a natural isomorphism between V and V** given by the function \omega^\epsilon which maps a vector in V to a set of numbers in V*. However, this isomorphism is not natural between V and V*. It is also important to note that this isomorphism is specific to the bilinear mapping \omega(x,f) = f(x), and may be confusingly stated as a set of numbers instead of a linear map.
  • #1
yifli
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natural isomorphism from V to V**

It is known that there is a natural isomorphism [tex]\epsilon \rightleftharpoons \omega^\epsilon[/tex] from V to V**, where [tex]\omega: V \times V* \rightarrow R[/tex] is a bilinear mapping.

So given a certain [tex]\epsilon \in V[/tex], its image under the isomorphism is actually a set of values [tex]\left\{f(\epsilon),f \in V^*\right\}[/tex], i.e., a vector is mapped to a set of numbers

Is my understanding correct?

Thanks
 
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  • #2


yifli said:
It is known that there is a natural isomorphism [tex]\epsilon \rightleftharpoons \omega^\epsilon[/tex] from V to V**, where [tex]\omega: V \times V* \rightarrow R[/tex] is a bilinear mapping.

So given a certain [tex]\epsilon \in V[/tex], its image under the isomorphism is actually a set of values [tex]\left\{f(\epsilon),f \in V^*\right\}[/tex], i.e., a vector is mapped to a set of numbers

Is my understanding correct?

Thanks

There is no natural isomorphism between V and V*. What isomorphism are you thinking of?

Generally, any isomorphism requires the choice of an inner product where a vector in V is identified with a linear map from V into the base field. You can think of a linear map as a set of numbers but it is more than that because it is not just any map but is a linear map.
 
  • #3


You are misreading what he said. There is no natural isomorphism from V to V*, the dual space, (if V has infinite dimension) but there is from V to V**, the dual of the dual.
 
  • #4


lavinia said:
There is no natural isomorphism between V and V*. What isomorphism are you thinking of?
it 's true that there's no natural isomorphism between V and V*, but I'm talking about the natural isomorphism between V and V**
 
  • #5


yifli said:
So given a certain [tex]\epsilon \in V[/tex], its image under the isomorphism is actually a set of values [tex]\left\{f(\epsilon),f \in V^*\right\}[/tex], i.e., a vector is mapped to a set of numbers

Is my understanding correct?
Not really. The image of [itex]\epsilon[/itex] is the function that takes [itex]f \in V^*[/itex] and maps it to [itex]f(\epsilon)[/itex]. I.E. [itex]\omega^\epsilon[/itex] the function defined by
[tex]\omega^\epsilon(f) = f(\epsilon)[/tex]​

Incidentally, I assume by [itex]\omega[/itex] you mean not a general bilinear mapping, but instead the specific map [itex]\omega(x,f) = f(x)[/tex]...

(Although, I could imagine what you wrote being intended to mean this, but stated awkwardly)
 

What is natural isomorphism from V to V?

Natural isomorphism from V to V refers to a bijective linear transformation from a vector space V to itself. This means that the transformation preserves the structure and properties of the vector space, and every element in V has a unique image under the transformation.

How is natural isomorphism different from regular isomorphism?

Natural isomorphism is a special type of isomorphism that is defined between a vector space and itself. Regular isomorphism, on the other hand, can be defined between any two structures, such as groups or rings. Additionally, natural isomorphism is uniquely determined by the vector space structure, while regular isomorphism can be defined in multiple ways.

What are the benefits of natural isomorphism?

Natural isomorphism allows us to translate results and properties from one vector space to another in a more efficient and intuitive way. It also helps us identify and understand the connections between different structures and their properties.

Can a vector space have multiple natural isomorphisms?

No, a vector space can only have one natural isomorphism to itself. This is because the definition of natural isomorphism requires the transformation to preserve the structure and properties of the vector space, and any other transformation would change these properties.

How is natural isomorphism related to the concept of duality?

Natural isomorphism is closely related to the concept of duality in mathematics. Duality is a relationship between two structures that allows us to switch between them in a natural way. Natural isomorphisms often arise in the context of duality, as they provide a way to relate and understand the properties of dual structures.

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