Extend Isometry from Semi-normed to Normed Space

In summary, the conversation discusses the extension of an isometry on a dense set of a semi-normed space to a unique linear isometry of the entire space. It is possible due to the fact that any element in the dense set can be represented as a limit of a Cauchy sequence, which can then be mapped to a Cauchy sequence in the normed space. This requires the normed space to be complete and Hausdorff in order to guarantee a unique limit and define the element as F(h).
  • #1
wayneckm
68
0
Hello all,


May someone help me on this question:

Suppose the map [tex] F [/tex] is an isometry which maps a dense set [tex] H [/tex] of a semi-normed space [tex] \mathcal{H} [/tex] to a normed space [tex] \mathcal{G} [/tex], now the theorem said we can extend this isometry in a unique manner to a linear isometry of the semi-normed space [tex] \mathcal{H} [/tex].

So I do not understand how and why can we do so?

Thanks very much!


Wayne
 
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  • #2
Is it because, due to the dense set, any element [tex] h [/tex] in [tex] \mathcal{H} [/tex] is the limit of a sequence of elements [tex] \{h_{n}\}\ [/tex] in [tex] H [/tex], so this forms a Cauchy sequence, and then, under the isometric map, we can obtain a Cauchy sequence in [tex] \mathcal{G} [/tex], however, I think we should assume the space [tex] \mathcal{G} [/tex] is complete and Hausdorff so that we are guaranteed there exists a unique limit, and so we can define such element as [tex] F(h) [/tex] ?
Am I correct? Thanks.
 

FAQ: Extend Isometry from Semi-normed to Normed Space

What is an isometry?

An isometry is a type of function that preserves distances between points in a space. In other words, if you apply an isometry to a shape or object, the resulting shape or object will have the same size and shape, but may be rotated or reflected.

What is a semi-normed space?

A semi-normed space is a vector space equipped with a semi-norm, which is a function that assigns a non-negative value to each vector in the space. However, unlike a norm, a semi-norm may assign a value of 0 to a non-zero vector.

Why is it important to extend isometry from semi-normed to normed space?

Extending isometry from semi-normed to normed space allows for a more accurate representation of the distances between points in a vector space. In normed spaces, the distance between two vectors is defined as the norm of their difference, which takes into account the magnitude of the vectors. This can provide more precise measurements and calculations in various fields such as physics, engineering, and computer science.

How is isometry extended from semi-normed to normed space?

To extend isometry from semi-normed to normed space, we must show that the function preserves the distances between points in the semi-normed space and the normed space. This can be done by proving that the semi-norm is equivalent to the norm in the normed space, meaning they assign the same value to any given vector. This can be shown using the triangle inequality and other properties of norms.

Are there any limitations to extending isometry from semi-normed to normed space?

There are some limitations to extending isometry from semi-normed to normed space. For example, not all semi-norms can be extended to norms. Additionally, the extension may not always be unique, meaning there may be multiple norms that can be extended from a given semi-norm. However, in most cases, it is possible to extend isometry from semi-normed to normed space with proper analysis and proof.

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