Dear Geoff,
I can't answer positively or not positively to your question( I don't have arguments), but I think that a quasi-Banach space of type 1 is not necessary equal to its Mackey topology completion.
I add some known facts regarding Lorentz spaces, maybe they are useful:
If q
As I remember, if we have a topological vector space (X,T) - T the linear topology on X, then a linear topology U on X is Mackey topology iff all linear and continuous functionals on X relatively to T are also continuous relatively to U. In other words, a Mackey topology is the maximal topology on X, preserving the continuous dual of (X,T). And, if (X,T) is locally convex( T is generated by the family of Minkowski functionals on X), then T is right Mackey topology on X.
Please correct me, if I'm wrong!
Now let's consider the Lorentz space L1,q. If 1/q+1/r=1 and L1,r may be identified with the dual of L1,q, I think this fact would help to obtain an answer to your question. But I don't know anything about duals of Lorentz spaces and my assertion is probably not true.
The notion of Mackey topology make sens, specially in separated locally convex space framework. If (E, T) is such a space with continuous dual E', the associated Mackey topology is the finest locally convex topology on E having E' again as continuous dual. A space whose topology is the corresponding Mackey topology is called a Mackey space. Any bornological (then also any metrizable and any normed) space is a Mackey space.
In the non locally convex space framework, if we adopt the same definition for the Mackey topology, then the latter may be "trivial". For instance the Mackey topology of l^p, for 1 < p
In case of non-locally-convex spaces having a non-trivial dual, "Mackey topology" may not be the best term but one could consider the non-Hausdorff 1-convex envelope, i.e. the non-Hausdorff vector-topology induced by the closed convex hull of each open set from the original topology. So, if Z is the subspace of L1,inf, L1,inf/Z is a non-trivial normed space. I don't believe this space is complete and hence L1,inf has a non-trivial, non-Hausdorff 1-convex envelope or "semi-Banach" envelope. However, I am not sure how to show this or if it is indeed true.
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