Question

If p is a sublinear functional on real vector space x show that there exist linear functional on x such that

Answer 1

Step-by-step explanation:

The Hahn-Banach Theorem is described as this:

X be a real vector space and p a sublinear functional on X. Furthermore, let f be a linear functional which is defined on a subspace Z of X and satisfies

f(x)≤p(x) for all x∈Z.

Then f could be extended on the whole X.

Now is the question, I cannot understand the following application of Hahn-Banach theorem:

Let X be a normed space and let x0≠0 by any element of X. Then there exists a bounded linear functional f~ such that

||f~||=1, f~(x0)=||x0||.

The proof is stated as following:

We consider the subspace of Z of X consisting of all elements x=αx0 where α is a scalar. On Z we define a linear functional f by

f(x)=f(αx0)=α||x0||.

f is bounded and has norm ||f||=1 because

|f(x)|=|f(αx0)|=|α|||x0||=||αx0||=||x||.

Then based on some extension of Hahn-Banach Theorem, f has a linear extension f~ from Z to X fulfill the condition.

I think I do not fully understand Hahn-Banach theorem. From my understanding, the functional f in the proof is not linear functional.

Let x=(−1+1)x0, then f(x) = f(0) = ||0|| = 0 instead of f(−x0+x0)=f(−x0)+f(x0)=||x0||+||x0||=2||x0||.

Where I got wrong?