Are functionals united with the vector space which they operate on?

In summary, Jake believes that everything can be represented with math, but that there are some things that can't be defined with math.
  • #1
jaketodd
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Are functionals united with the vector space which they operate on? For example, Physics is a functional of Behavioral Psychology. However, Behavioral Psychology does not include Physics. Am I correct?

Thank you,

Jake
 
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  • #2
Is this a joke?
 
  • #3
jaketodd said:
Are functionals united with the vector space which they operate on? For example, Physics is a functional of Behavioral Psychology. However, Behavioral Psychology does not include Physics. Am I correct?

Thank you,

Jake
What does this have to do with "vector spaces"? For that matter what does it have to do with mathematics?

Perhaps it would help if you gave a precise definion for your use of "functional" here.
 
  • #4
I assumed (apparently incorrectly) that functionals could be applied to the abstraction levels of concepts.

Sorry about the confusion; I'm new to these concepts.

What got me asking this question is from Wikipedia: "A spin network, immersed into a manifold, can be used to define a functional on the space of connections on this manifold."
http://en.wikipedia.org/wiki/Spin_network

So I guess what I'm really asking is: Is the map the functional provides, on the space of connections on a manifold, united with the spin network that is immersed into the manifold in order to obtain the functional?

Thanks,

Jake
 
  • #5
Also, could you give us a mathematical meaning to "united"?
 
  • #6
zhentil said:
Also, could you give us a mathematical meaning to "united"?

The best I can do is try conceptually: In the concept of spacetime, space is united with time. Does that bring to mind a mathematical representation?

Thank you for bearing with me,

Jake
 
  • #7
Lol, this is really funny actually. No offense, but you're not making any sense. :) . These math concepts have very precise meanings... if you want a philosophical discussion, you should ask in the philosophy forum. :)
 
  • #8
But isn't everything representable with math?
 
  • #9
No, why in the world would you think so?
 
  • #10
What can't be defined with math? That's kind of a philosophy question, and I can already hear objections to this thread turning into that. So if you don't want to, don't worry about not responding.
 

FAQ: Are functionals united with the vector space which they operate on?

1. What is a functional in the context of vector spaces?

A functional is a mathematical object that takes in a vector as input and returns a scalar value as output. It can be thought of as a function that operates on a vector space, hence the name "functional."

2. How are functionals related to vector spaces?

Functionals are closely related to vector spaces because they operate on them. A functional is usually defined as a linear map from a vector space to its underlying field of scalars. This means that functionals are intimately tied to the structure and properties of the vector space they operate on.

3. Can functionals be united with any vector space?

Yes, functionals can be defined for any vector space as long as the underlying field of scalars is well-defined. This means that functionals can be united with vector spaces over real or complex numbers, as well as more abstract vector spaces like function spaces.

4. What is the importance of functionals in mathematics?

Functionals play a crucial role in many areas of mathematics, including functional analysis, differential equations, and optimization. They provide a powerful tool for studying and understanding the behavior of vector spaces, as well as for solving various mathematical problems.

5. How are functionals used in real-world applications?

Functionals have numerous applications in real-world problems, particularly in physics and engineering. For example, functionals are used in the calculus of variations to find the optimal solution to a physical system, and in control theory to design optimal control systems. They are also used in computer graphics and image processing to analyze and manipulate data.

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