Math Forum Discussions - Re: Which TVS is compatible with Lie bracket?

Anton Deitmar

Posts: 97
Registered: 1/25/05

Re: Which TVS is compatible with Lie bracket?
Posted: Sep 26, 2005 2:30 PM

Plain Text

> in the following V is a linear space and T is a
> linear map from V to V
>
> Question 1:Is there a norm on V such that T is a
> bounded operator

Counterexample: let V have the basis e_1, e_2, e_3, ...
Define T by

T(e_j) = j e_j

Then for any norm N:

N(T(e_j)) = j N(e_j)

so T is not bounded.

>
> question 2:is there a topological vector space
> structure on V such
> that T is continious linear map?

Yes, let N be a norm on V and set

s_k(v) = N(T^k(v))

Then the family s_k, k=0,1,2,...
of seminorms defines a topology on V and one has

s_k(T(v)) = s_{k+1}(v),

hence T is continuous wrt this topology.

Cheers,
Anton