 
Summary: Tangency.
Let X be a normed vector space.
Definition. Suppose v X and C X. We say C is a cone with vertex v if
x C {v} and t 0 v + t(x  v) C.
Note that the empty set is a cone with vertex v and that v C if C {v} = .
Proposition. Suppose v X and C is a nonempty family of cones with vertex v. Then C is a cone with
vertex v.
Proof. This is immediate.
Proposition. Suppose v X and C is a cone with vertex v. Then the closure of C is a cone with vertex
v.
Proof. Exercise.
Definition. Suppose A X, a acc A. For each > 0 we let
Tana(A, ) = cl {t(x  a) : t 0, and x (A {a}) Ba()}.
Note that. by virtue of the previous Proposition, Tana(A, ) is a closed cone with vertex 0.
We let
Tana(A) =
>0
Tana(A, )
and we let
Nora(A) = { X
