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!

The answer to the first question is clearly _no_ , since the mapping can collapse the domain to the zero vector. A function that simply interchanges two points has an inverse that takes bounded sets to bounded sets, but the function is neither continuous nor linear.

The second statement is precisely equivalent to the first, so it does indeed mean that the image of a bounded set under a continuous linear mapping is bounded. The answer to the final question is _no_ , just as in the first part.