[Math] Under what conditions a bounded linear map can be extended

Question

I have two questions after reading the Hahn-Banach theorem from Conway's book ( I have googled to know the answer but I have not found any result yet. Also I am not sure that whether my questions have been asked here somewhere on this forum – so please feel free to delete them if they are not appropriate )

Here are my questions:

1. We know that if $M$ is a linear subspace of $X$ and $f :M\to\mathbb{F}$ and $f$ is linear,bounded by a seminorm $p$ then $f$ can be extended onto $X$ by some functional $F$. Can $F$ be unique ? Under what condition $F$ will be an unique extension? It would be appreciate if you could give me one example that $F$ could not be unique.

2. If the above $\mathbb{F}$ is replaced a Banach space $Y$, i.e, let $M$ be a closed subspace of a Banach space $X$, and $f :M\to Y$ be a bounded linear operator, can we extend $f$ by a bounded operator $F :X\to Y$ ? if not, what condition should be put on $Y$ to have a such extension?

thanks so much

Answer 1

$Y$ is called an injective Banach space if the extension exists for all $X$, $M$, and $f$. An example is $Y = l^\infty$. (Should be in Banach space text books. Here's a paper: http://www.jstor.org/pss/1998210 )