Isomorphism homework help

In summary: The difference of two members is just in the infinity-norm. How can you see that it's decreasing? (the only useful thing you can do is the triangle inequality, right?)
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  • #2


Tf = f'. That is, the operator applied to f, in this case a differential operator. T is bounded because ||Tf|| = ||f||. [tex]T^{-1}[/tex] is defined to be [tex]T^{-1}f' = f[/tex] and so is also bounded by the same reason. This comes from the definition of boundedness of linear operators.
 
  • #3


cellotim said:
Tf = f'. That is, the operator applied to f, in this case a differential operator.
Ok, that's clear.

T is bounded because ||Tf|| = ||f||. [tex]T^{-1}[/tex] is defined to be [tex]T^{-1}f' = f[/tex] and so is also bounded by the same reason. This comes from the definition of boundedness of linear operators.
I don't see how you prove that it is bounded. Why is||Tf|| = ||f||. [tex]T^{-1}[/tex] ?
 
  • #4


The definition of boundedness of linear operators on normed spaces:

[tex] ||Tf||_\infty \leq M||f||_E [/tex] for some M. If we let M=1, then we have the proof for T and its inverse..
 
  • #5


OK clear. But how can I compute its inverse? I know it has to go from [tex]||f'||_{\infty}[/tex] to f,right?
 
  • #6


You don't need to compute anything. You know that [tex]T^{-1}f' = f[/tex], that's all you need.
 
  • #7
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  • #8


You need to show that given a sequence [tex]f_n\in E[/tex], where [tex]||f_m - f_n||_E < 1/k[/tex] for any m, n>N, some N, and any k, that its limit is in E, i.e. that [tex]lim_{n\rightarrow\infty} f_n = f[/tex] such that f(0) = 0 and f is continuously differentiable on the interval [0,1]. The first part is easy. To prove C1, the key is in the norm. The norm is the sup of the derivative meaning that sup of the derivative of the difference of two members of the sequence becomes smaller. This keeps the derivative of f from exploding.
 
  • #9


cellotim said:
You need to show that given a sequence [tex]f_n\in E[/tex], where [tex]||f_m - f_n||_E < 1/k[/tex] for any m, n>N, some N, and any k, that its limit is in E, i.e. that [tex]lim_{n\rightarrow\infty} f_n = f[/tex] such that f(0) = 0 and f is continuously differentiable on the interval [0,1].

What I want to show is that [tex] ||f - f_n ||_E < \epsilon [/tex] for n>N is that the same?

The first part is easy. To prove C1, the key is in the norm. The norm is the sup of the derivative meaning that sup of the derivative of the difference of two members of the sequence becomes smaller. This keeps the derivative of f from exploding.

I don't understand: the difference of two members is just in the infinity-norm. How can you see that it's decreasing? (the only useful thing you can do is the triangle inequality, right?
 

What is isomorphism?

Isomorphism refers to a mathematical concept in which two objects or structures are essentially the same, despite having different forms or appearances. In other words, they can be transformed into each other without losing any important characteristics or properties.

What is isomorphism used for?

Isomorphism has many applications in various fields of science, including chemistry, biology, and computer science. It is particularly useful in understanding the relationships between different structures or systems, and in identifying patterns and similarities between seemingly unrelated objects.

How is isomorphism different from homomorphism?

While both isomorphism and homomorphism involve structural similarities between objects, they have different levels of equivalence. Isomorphism means two objects are exactly the same, while homomorphism means they are similar in some way, but not necessarily identical.

What are some examples of isomorphism?

Some common examples of isomorphism include the isomorphism between the human hand and the paw of a cat, the isomorphism between DNA and RNA molecules, and the isomorphism between different chemical compounds with the same molecular formula.

How do I solve problems involving isomorphism?

To solve problems involving isomorphism, you need to first identify the structures or objects in question and determine if they are indeed isomorphic. This can be done by comparing their properties, functions, and structural features. Then, you can use this information to make predictions or solve equations involving these structures.

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