The Hahn-Banach Theorem Review

The Hahn-Banach Theorem Review

We will now review some of the recent material regarding the Hahn-Banach theorem.

(1)
\begin{align} \quad p(\lambda x) = \lambda p(x) \end{align}
  • Furthermore, a function $p : X \to [0, \infty)$ is said to be Subadditive if for all $x, y \in X$ we have that:
(2)
\begin{align} \quad p(x + y) \leq p(x) + p(y) \end{align}
  • On The Hahn-Banach Lemma page we proved that if $X$ is a linear space, $Y \subset X$ is a subspace, and $p : X \to [0, \infty)$ is a positively homogeneous and subadditive function, then if $\varphi : Y \to \mathbb{R}$ is an $\mathbb{R}$-linear functional such that $\varphi(y) \leq p(y)$ for all $y \in Y$ then for each $z \in X \setminus Y$ there is an $\mathbb{R}$-linear functional $\Phi : Y \oplus \mathrm{span} (z) \to \mathbb{R}$ such that:
(3)
\begin{align} \quad \Phi(y) &= \varphi(y), \quad \forall y \in Y \\ \quad \Phi(x) &\leq p(x), \quad \forall x \in Y \oplus \mathrm{span} (z) \end{align}
  • We then proved the Hahn-Banach theorem on The Hahn-Banach Theorem (Real Version) and The Hahn-Banach Theorem (Complex Version) pages. We proved that if $X$ is a linear space, $Y \subset X$ is a subspace, and $p : X \to [0, \infty)$ is a subadditive function such that $p(\lambda x) = |\lambda|p(x)$ for all $\lambda \in \mathbb{C}$ and for all $x \in X$ then if $\varphi : Y \to \mathbb{C}$ is a linear functional such that $|\varphi(y)| \leq p(y)$ for all $y \in Y$ then there exists a linear functional $\Phi : X \to \mathbb{C}$ such that:
(4)
\begin{align} \quad \Phi(y) = \varphi(y), \quad \forall y \in Y \\ \quad |\Phi(x)| \leq p(x), \quad \forall x \in X \end{align}
  • In other words, the Hahn-Banach theorem tells us that whenever we have a linear functional $\varphi$ defined on a subspace of $Y$ of $X$ that is dominated by a function $p$ with the properties above, then we can always find a linear functional extension $\Phi$ of $\varphi$ defined on the whole space $X$, that is dominated by $p$ on the whole space $X$.
  • We then looked at some important applications of the Hahn-Banach theorem which are summarized in the table below.
Page Result
Extensions of Linear Functionals with Equal Norms Let $X$ be a normed linear space and let $Y \subset X$ be a subspace. If $\varphi \in Y^*$ then there exists a continuous linear functional $\Phi \in X^*$ such that $\Phi(y) = \varphi(y)$ for all $y \in Y$ and $\| \Phi \| = \| \varphi \|$.
Extensions of Linear Functionals with Equal Norms Let $X$ be a normed linear space and let $Y \subset X$ be a subspace. For every $x_0 \in X$ there exists a continuous linear functional $\Phi \in X^*$ such that $\Phi(x_0) = \| x_0 \|$ and $\| \Phi \| = 1$.
Finite-Dimensional Subspaces of Normed Linear Spaces have Topological Complements Let $X$ be a a normed linear space and let $Y \subseteq X$ be a subspace. If $Y$ is finite-dimensional then $Y$ has a topological complement.
Criterion for a Point to be in the Closure of a Subspaces of Normed Linear Spaces Let $X$ be a normed linear space and let $Y \subseteq X$ be a subspace. Then $x_0 \in X$ is contained in the closure of $\overline{Y}$ of $Y$ if and only if whenever $\varphi \in X^*$ is such that $\varphi(y) = 0$ for all $y \in Y$ then $\varphi(x_0) = 0$.
The Canonical Embedding J is an Isometry Let $X$ be a normed linear space. Then the canonical embedding $J : X \to X^{**}$ is an isometry.
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