Does a linear map of normed vector spaces have a unique extension? Let $ V$ 
  and $ W$ 
  be finite dimensional normed vector spaces and let $ U$ 
  be an open subset of $ V$ 
  (the topology induced by the norms). Let $ L:U\rightarrow W$ 
  be a linear map. Does there exist a unique linear map $ T:V\rightarrow W$ 
  such that the restriction of $  T$ 
  to $ U$ 
  is $ L$ 
 ? If this is not true, then are there any additional conditions which can be imposed on $ V$ 
  and $ W$ 
  such that the extension exists? In particular, I am interested in the case where $ V$ 
  and $ W$ 
  are Euclidean.
The domain $U$ of the linear map $L$ shoud be a subspace of $V$ !!
If $U$ is a non- empty open subset of $V$ and is a subspace of $V$, then we have $U=V$ ! Try a proof
The domain $U$ of the linear map $L$ shoud be a subspace of $V$ !!
If $U$ is a non- empty open subset of $V$ and is a subspace of $V$, then we have $U=V$ ! Try a proof