Discontinuous linear mapping between infinite-dimension vector space

In summary, the standard example of a linear mapping between two infinite dimensional normed vector space that is discontinuous is differentiation on the set of all real polynomials with the sup-norm as the norm. This operator is not bounded and has negative consequences.
  • #1
yifli
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It is known that any linear mapping between two finite dimensional normed vector space is continuous (bounded).

Can anyone give me an example of a linear mapping between two infinite dimensional normed vector space that is discontinuous?

Thanks
 
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  • #2
Hi yifli! :smile:

yifli said:
It is known that any linear mapping between two finite dimensional normed vector space is continuous (bounded).

Can anyone give me an example of a linear mapping between two infinite dimensional normed vector space that is discontinuous?

Thanks

The standard example is a very familiar linear mapping: differentiation. Let [itex]X[/itex] be the set of all real polynomials on [0,1]. Equip this with the sup-norm, i.e.

[tex]\|f\|_\infty=\sup_{t\in [0,1]}{|f(t)|}[/tex]

Let

[tex]T:X\rightarrow X:f\rightarrow f^\prime[/tex]

Let [itex]p_n(x)=x^n[/itex], then [itex]\|p_n\|_\infty=1[/itex], but

[itex]\|T(p_n)\|_\infty=n\|p_n\|_\infty[/itex]

thus the operator T is not bounded.

This is a very tragic result and has a lot of bad consequences...
 

FAQ: Discontinuous linear mapping between infinite-dimension vector space

1. What is a discontinuous linear mapping between infinite-dimension vector spaces?

A discontinuous linear mapping between infinite-dimension vector spaces is a function that maps vectors from one infinite-dimensional vector space to another, but is not continuous. This means that small changes in the input vector can result in large changes in the output vector, making the function unpredictable and difficult to analyze.

2. How is a discontinuous linear mapping different from a continuous linear mapping?

A continuous linear mapping is a function that preserves the structure of the vector space, meaning that small changes in the input vector result in small changes in the output vector. A discontinuous linear mapping, on the other hand, does not preserve this structure and can have unpredictable behavior.

3. What are some applications of discontinuous linear mappings in mathematics?

Discontinuous linear mappings are often used in functional analysis, a branch of mathematics that studies infinite-dimensional vector spaces and their linear transformations. They can also be used in nonlinear dynamics and chaos theory to model complex systems.

4. Can a discontinuous linear mapping be invertible?

No, a discontinuous linear mapping cannot be invertible because it does not preserve the structure of the vector space. This means that there is no way to accurately reverse its effects and retrieve the original input vector.

5. What are some challenges in studying discontinuous linear mappings?

One of the main challenges in studying discontinuous linear mappings is that they do not follow the same rules and properties as continuous linear mappings. This can make it difficult to apply traditional mathematical techniques and requires specialized methods for analysis. Additionally, the unpredictable behavior of discontinuous linear mappings can make them difficult to understand and model.

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