such that .
linear subspace, , belongs to .
is a seminorm on with .
By the Hahn-Banach theorem, linear form such that and .
and .
So we found with .
Well, if you have a (real or complex) linear space, then is called a seminorm if:
1) (i.e. is sublinear) and
2) (in or ).
The Hahn-Banach theorem in its complex form (which is a quick corollary of the initial Hahn-Banach theorem) states that if we have a complex linear space, a seminorm on , a linear subspace of , and a linear form on such that , then there exists a linear form on which is equal to on and which satisfies the inequality .
Let me know if you have any more questions
Thank you again that is very helpful and insightful.
I've got a better idea now but still struggling with one thing.
Its the way you define the mapping of Y in the second line
" alt="Y:=\mathbb{C}x_0\subset X
" />
It seems strange to me as \mathbb{C} is the symbol for the set of Complex numbers. Should this be written differently?
Please explain this notation if you dont mind.