Proving the Closedness of a Linear Subspace in a Normed Space Using Dual Spaces

[math8]

Let X be a normed space, F be a closed linear subspace of X,
Let z be in X, z is not in F.
Let S={x+az:a is in the field Phi}=Span of F and z
We show S is closed.

I would define a function f : S--> Phi by f(x+az)=a
and show that |f|< or equal to 1/d where d= distance(z,F)
hence f is in S* (dual of S).
So we let {x_n +a_n z} be a sequence converging to w (x_n is in F,a_n is in Phi). We show w is in S.
We note {f(x_n +a_n z)} is a cauchy sequence.

But I am not sure how to proceed.

 

[Dick]

You might also note that for x_n in F, d(x_n+a_n*z,F)=d(a_n*z,F). You want to show x_n and a_n*z are both convergent separately.