I am having trouble with a result in my text left as an exercise.(adsbygoogle = window.adsbygoogle || []).push({});

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?

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

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