Under what conditions a bounded linear map can be extended ? 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: 


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*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.

*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 
 A: $Y$ 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 )
A: Note quite what you asked, but related:
Continuous extensions of (continuous) functionals from $M$ are unique if and only if $M$ is a dense subspace of $X$. Otherwise its closure is a proper closed subspace and therefore there exists a nonzero bounded functional $\phi$ vanishing on the closure, which implies that $F+\phi$ is bounded and also extends $f$.
For the second question, it is easier to put conditions on $M$ so that for every $Y$, every map from $M$ to $Y$ can be extended. As mentioned in the comments, a necessary and sufficient condition is that $M$ is a complemented subspace of $X$.
A: When $X^*$ is strictly convex and $M$ is a subspace of $X$, every norm one $f\in M^*$ admits a unique norm one extension $F\in X^*$, because given two of such extensions $F_1$ and $F_2$, $(F_1+F_2)/2$ is norm one. 
In fact this property characterizes $X^*$ strictly convex. See Theorem 7.11 in  B.V. Limaye "Functional analysis", 2nd. ed.
