Homomorphic normed linear spaces

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In summary, the conversation discusses the concepts of homomorphic normed linear spaces and the difference between homomorphism and isomorphism. The first question asks if two finite-dimension homomorphic normed linear spaces must have the same dimension, and if a homomorphism is possible between an infinite-dimension and a finite-dimension normed space. The second question asks if a homomorphism between two normed linear spaces is always uniformly continuous. The conversation also touches on the use of the terms "homomorphic" and "isomorphic" and their relation to homeomorphism and isomorphism. The participants also discuss the difficulty in showing that two spaces are not homeomorphic and the tools needed for such proofs.
  • #1
Hi all experts,
I've just visitted another Maths forum and picked up two interesting questions:
Question 1. Let H and K be homomorphic normed linear spaces. Is it necessary that H and K have the same dimension if both H and K are finite-dimension ? Is there possible a homomorphism between an infinite-dimension normed space and a finite-dimension one ?
Question 2. If f is a homomorphism between two normed linear spaces, is f necessary uniformly continuous ?
Hope to get your ideas.
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  • #2
Let H and K be homomorphic normed linear spaces.
Did you menan to say "isomorphic"?

Have you given much thought to these questions?
  • #3
By saying "H and K is homomorphic", I mean that there is an one-to-one mapping f from H to K such that both f and f^{-1} (the inverse of f) are continous. Perhaps, this concept should be called "isomorphic" and sorry if I made a mistake.

If H=R and K=R^2, I think R and R^2 cannot be holomorphic because if we remove a point of R, such as x, then R\{x} is no longer a connected set, but R^2\{f(x)} is still a connected set. Here, f is an one-to-one from R to R^2 such that f and f^{-1} are continuos.
  • #4
A continuous bijection with a continuous inverse is an homeomorphism.
  • #5
A one-to-one homomorphism between 2 algebraic structures is an isomorphism.

  • #6
No it isn't. The inclusion of Z into Q or R (as additive groups) is a one to one homomorphism. It is not an isomorphism.
  • #7
So, I think the phrase needed here is "homeomorphic isomorphism" or perhaps "isomorphic homeomorphism"!
  • #8
To show R^n is not isomorphic to R^m as a vector space for n[itex]\neq[/itex]m, you can just show that isomorphism preserves dimension (ie, show a basis is mapped to a basis). Showing they are not homeomorphic as topological spaces is much more difficult. It can be done using cut points as in post 3 for n=1, but above this you need more advanced tools from algebraic topology. And what do you mean by uniformly continuous?
  • #9
real normed linear spaces of finite dimension are probably always isomorphic to R^n. then they are determined by their homeomorphism type i would guess. a proof tends to involve homology theory.

What is a homomorphic normed linear space?

A homomorphic normed linear space is a mathematical structure that combines the properties of a normed linear space and a homomorphism. It is a vector space equipped with a norm function and a homomorphism function that preserves the algebraic operations of addition and scalar multiplication.

How does a homomorphic normed linear space differ from a traditional normed linear space?

In a traditional normed linear space, the norm function only maps vectors to non-negative real numbers. In a homomorphic normed linear space, the norm function also maps vectors to the algebraic structure of the homomorphism, which can be a different set of numbers or objects.

What are some examples of homomorphic normed linear spaces?

One example is the space of continuous functions on a closed interval, where the norm function is the maximum absolute value of the function and the homomorphism function is the integration operation. Another example is the space of square-integrable functions, where the norm function is the square root of the integral of the function squared and the homomorphism function is the multiplication operation.

What are the applications of homomorphic normed linear spaces?

Homomorphic normed linear spaces have applications in functional analysis, operator theory, and signal processing. They are also useful in studying the convergence and stability of numerical algorithms and approximations.

What are the properties of homomorphic normed linear spaces?

Some important properties of homomorphic normed linear spaces include completeness, continuity, and linearity. They also have properties related to the norm and homomorphism functions, such as sub-multiplicativity, homogeneity, and preservation of algebraic operations.

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