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A TVS is seminormable if and only if it has a convex bounded neighborhood of the origin. Thus a locally convex TVS is seminormable if and only if it has a non-empty bounded open set.A TVS is normable if and only if it is a T₁ space and admits a bounded convex neighborhood of the origin.

If is a Hausdorff locally convex TVS then the following are equivalent:

1. is normable.
2. is seminormable.
3. has a bounded neighborhood of the origin.
4. The strong dual of is normable.
5. The strong dual of is metrizable.

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