# Continuous linear mapping and bounded subsets

1. Continuous linear mappings between topological vector spaces preserve boundedness.

I was wondering if it means that the inverse image of a bounded subset under a continuous linear mapping is still bounded?

Conversely, must a mapping between two topological vector spaces, such that the inverse image of any bounded subset is still bounded, be continuous linear?

2. A continuous linear operator maps bounded sets into bounded sets.

Does it mean that the image of a bounded subset under a continuous linear mapping is still bounded?

Conversely, must a mapping between two topological vector spaces that maps bounded sets to bounded sets be continuous linear?

Thanks and regards!

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The answer to the first question is clearly no, since the mapping can collapse the domain to the zero vector. –  Brian M. Scott Feb 25 '12 at 2:07
The answer to the second is yes. But the second is not true suppose for example a first space as a separable space and put ${x_{k}}$ a dense sequence then the function $f$ such that $f(x_{k})=1$ and zero case contrary, this function is neither continuous nor liner but naturally bounded. –  checkmath Feb 25 '12 at 2:37

@Tim: No, they’re fine; they simply don’t say anything about the inverses of continuous linear maps. They’re equivalent because they mean exactly the same thing: if $B$ is bounded, then $f[B]$ is bounded. –  Brian M. Scott Feb 25 '12 at 2:24