Re: equivalent norms

Arturo Magidin wrote:

In article <8d7b8$45116ba1$9161c4ca$21998@xxxxxxxxxxxxxxxx>,
Felix4 <f_felix4@xxxxxxxxxxxx> wrote:
Hi,

how to prove formally that if two norms on a vector space V are
equivalent, then their balls are contained in each other?

Question 1: What is your definition of "equivalent norms"? (I ask
because there are several ways of phrasing it, and the precise wording
of your definition will be relevant).

Two norms N1,N2 are equivalent if there exist positive reals a,b such that
for all x you have a*N1(x) <= N2(x) <= b*N1(x).

Question 2: What do you mean by "their balls are contained in each
other"? You have to be precise. Presumably, you mean that if you have
two norms, N1 and N2, then for any given point x and any given e>0,
there exist d1,d2>0 such that the N1-ball of radius e and centered at
x contains the N2-ball of radius d2 centered at x, AND the N2-ball of
radius e and centered at x contains the N1-ball of radius d1 and
centered at x.
Yes.
and is the converse true?

Same as point 1.

it seems ok on a picture, but i just cant find the formal argument

any help would be nice!

For example, Bourbaki defines a norm as a mapping
||.|| : V --> [0,oo) that satisfies:

NO1: ||x|| = 0 if and only if x = 0.
NO2: || x + y || <= ||x|| + ||y|| for all x,y in V.
NO3: ||tx || = |t| ||x|| for all scalars t and all x in V.

("General Topology", Chapter IX, Section 3.3). Then Bourbaki defines
two norms to be equivalent if and only if they define the same
topology. From that definition, the result will follow (formally) by
observing that the set of open balls centered at x form a basis for
the system of neighborhoods of x in the topology, and drawing the
conclusions.

On the other hand, others define equivalence in terms of certain
inequalities that must hold between the norm values. In which case, a
(formal) proof would require you to play with those inequalities to
show that if a certain point is in a sufficiently small ball centered
at x (relative to one norm) then it will lie in the desired ball
relative to the other norm.

So you need to say what your definitions actually are.


.

Relevant Pages

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