Extension of Lipschitz Continuous Operators on arbitrary sets in Hilbert Spaces Let $T: D(T) \subset X \to Y $ be a Lipschitz continuous operator on an arbitrary set $D(T)$ in the Hilbert Space $X$. Show that $T$ can be extended to an operator $\tilde{T}:X \to Y$ which is Lipschitz on the whole $X$ . 
I have no idea on how to proceed. Any hints will be appreciated. 
Thanks for the help!!
 A: First you need to extend the operator to the vector subspace $V \subset X$ spanned by $D(T)$. Once that's done the Hahn-Banach theorem should get you the rest of the way.
I'm not sure what an operator on an arbitrary set it. Usually operators are only defined on vector spaces. Let's ignore that for the moment.
We can write $V $ as $ \{c_1 a_1 + \ldots + c_n a_n \colon $ each $c_i \in \mathbb K $ and $ a_i \in D(T)\}$. 
So, being linear, the extension $T' \colon V \to Y$ would have to satisfy the relation . . . 
$ T'(c_1 a_1 + \ldots + c_n a_n)= c_1 T(a_1) + \ldots + c_n T(a_n)$
. . . if it exists at all. But trying to define $T'$ like this may lead to it not being well-defined, since two elements of $V$ might have different expressions in terms of the $a_i$.
To remedy this I suggest you choose a basis for $V$ out of the elements of $D(T)$ and try to define $T'$ as above, but under the condition that $c_1 a_1 + \ldots + c_n a_n$ is the unique expression of that element as a finite sum of basis elements.
