The Hahn-Banach Theorem

Theorem 9 (Hahn-Banach theorem)

Let \(X\) be a normed space, let \(Y \subset X\) be a subspace. Then any continuous linear functional \(\lambda \in Y^*\) on \(Y\) can be extended to a continuous linear functional \(\hat{\lambda} \in X^*\) on \(X\) with the same operator norm, thus \(\hat{\lambda}\) agrees with \(\lambda\) on \(Y\) and \(|| \hat{\lambda}||_{X^*} = ||\lambda||_{Y^*}\). Note that \(\hat{\lambda}\) is not necessarily unique.

Geometric Implications

There are three illustrative corollaries of the Hahn-Banach theorem.

Corollary 3 (The distinguishing property)

Let \(x, y \in X\). If \(f(x) = f(y)\) for all \(f \in X^*\) then \(x = y\).

Corollary 4 (Norming property)

For each \(x \in X\) there exists \(f \in X^*\) with \(f(x) = ||x||\) and \(||f||_{op} = 1\).

Corollary 5 (Classification of closures)

Let \(M\) be a linear subspace of a normed space \(X\) and let \(x_0 \in X\) then \(x_0 \in \overline{M}\) if and only if there exists no bounded linear functional \(f\) such that \(f(x) = 0\) for all \(x \in M\) but \(f(x_0) \neq 0\).