Question

A mapping from normed space x in to normed space y is operate​

Answer 1

Let A:U→V be bounded. If X⊂U is bounded there exists a K∈R such that ||x||<K for all x∈X. Let y∈A(X) i.e. Ax=y for an x∈X then

||y||=||Ax||≤||A||⋅||x||≤||A||⋅K

therefore A(X) is bounded too.

Now let A send bounded sets to bounded sets and let K be a bound for the image of the unit ball under A then

||A||=sup||x||≤1||Ax||≤K, i.e. A is bounded.

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