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  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 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.2012-02-25

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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.

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    @Tim: Unfortunately, that statement is a bit misleading. It would be better to say that morphisms **respect** the structure, meaning that their behavior is in some important way related to the structure. They don’t in general preserve all aspects of structure: when they do, they are isomorphisms in linear algebra and group theory and homeomorphisms in topology. In particular, a continuous function definitely does **not** necessarily preserve openness, though it does preserve other aspects of topological structure (e.g., convergence of sequences).2012-02-25