Relationship between seminormed, normed, spaces and Kolmogrov top. spaces

In summary, the text claims that a topological space X is normed if and only if it is Kolmogrov. However, the text does not provide a clear explanation of how to prove this. The first direction of the problem is correct, but the second direction is not clear.
  • #1
Ocifer
32
0
I am having trouble with a result in my text left as an exercise.

Let (X, τ) be a semi-normed topological space:

norm(0) = 0
norm(a * x) = abs(a) * norm(x)
norm( x + y) <= norm(x) + norm(y)

My text states that X is a normed vector space if and only if X is Kolmogrov. It claims it to be trivial and leaves it as an exercise. I'm able to get one direction, but I'm not very happy with it.

------------------------------------------------------------
(=> direction)

Assume X is normed, let x,y be in X, such that x != y.

Then d(x,y) > 0. I am tempted to use an open ball argument, claiming that there exists an open ball about x which cannot contain y, but then how do I relate this notion of open balls in a metric space to the open set required by the Kolmogrov (distinguishable) condition?

-------------------------------------------------------------------------------

In terms of the other direction, I am entirely lost (Kolmogrov and semi-normed imples normed).

Can anyone provide some insight, I just took a course in Real Analysis last semester and I did quite well, and I'm having trouble generalizing my insights now to topological spaces. With the semi-normed space we lose discernability, and I have a hunch that the Kolmogrov condition patches that problem, but I just can't get there.
 
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  • #2
For the => direction you are on the right track. You want an open ball B with center x, but with radius small enough that y is not in B. d(x,y) seems to be the only number you have to work with so try choosing a radius based on that. If you can find a radius such that y is not in B, then you are done because open balls are open sets.

In the other direction you wish to show [itex]\|x\| = 0[/itex] implies x=0. Suppose [itex]x \not= 0[/itex], then by the Kolmogorov condition you can find an open neighborhood U of either x or 0 which does not contain the other. Try to use this to find a radius r such that B(x,r) or B(0,r) is contained in U, in which case you can show [itex]\|x\|\not=0[/itex].

EDIT: Here B(p,r) means the open ball with center p and radius r.
 
  • #3
Thank you for your reply.
 

Related to Relationship between seminormed, normed, spaces and Kolmogrov top. spaces

1. What is the difference between seminormed and normed spaces?

Seminormed spaces are vector spaces with a function that assigns a non-negative value to each vector, while normed spaces are vector spaces with a function that assigns a positive value to each vector. In normed spaces, the function must also satisfy the triangle inequality, while in seminormed spaces it may not.

2. How are seminormed and normed spaces related to each other?

Seminormed spaces are a generalization of normed spaces. This means that all normed spaces are also seminormed spaces, but not all seminormed spaces are normed spaces.

3. What is the significance of Kolmogrov topological spaces in relation to seminormed and normed spaces?

Kolmogrov topological spaces are a generalization of both seminormed and normed spaces. They allow for the consideration of infinite-dimensional spaces and provide a framework for studying the continuity and convergence of functions in these spaces.

4. Can seminormed and normed spaces be used interchangeably in mathematical proofs?

No, while they are related, seminormed and normed spaces have distinct properties that make them suitable for different types of proofs. For example, normed spaces are better suited for proving convergence of sequences, while seminormed spaces are useful for studying partial derivatives.

5. How do seminormed and normed spaces relate to the concept of completeness?

A normed space is complete if every Cauchy sequence in the space converges to a point within the space. Similarly, a seminormed space is complete if every Cauchy sequence in the space converges to a point within the space's completion. This highlights the importance of completeness in both types of spaces.

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