3.1 Statement and proof of the Hahn-Banach Theorem

The Hahn-Banach theorem is a cornerstone of functional analysis. It allows us to extend bounded linear functionals from subspaces to entire normed spaces, keeping their norms intact. This powerful tool opens doors to studying dual spaces and weak topologies.

The theorem's proof uses Zorn's lemma, a heavy-duty tool in set theory. It constructs a maximal extension of the original functional, showing that this extension must cover the whole space. This approach gives us the extension's existence without explicitly building it.

Hahn-Banach Theorem

Hahn-Banach theorem for normed spaces

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• Assumes $X$ is a normed linear space contains a subspace $Y$
• Considers a bounded linear functional $f$ defined on the subspace $Y$
• Guarantees the existence of a bounded linear functional $F$ defined on the entire space $X$ that extends $f$
  • Preserves the values of $f$ on $Y$, so $F(y) = f(y)$ for all $y \in Y$
  • Maintains the norm of $f$, ensuring $\|F\| = \|f\|$

Proof using Zorn's lemma

• Defines the set $\mathcal{F}$ consisting of pairs $(Z, g)$
  • $Z$ represents a subspace of $X$ containing the original subspace $Y$
  • $g$ is a bounded linear functional on $Z$ that extends the original functional $f$
• Introduces a partial order on $\mathcal{F}$ where $(Z_1, g_1) \leq (Z_2, g_2)$ if $Z_1$ is a subset of $Z_2$ and $g_2$ restricted to $Z_1$ equals $g_1$
• Demonstrates that every chain in $\mathcal{F}$ has an upper bound
  • Constructs the upper bound $(Z, g)$ by taking the union of subspaces $Z_\alpha$ and defining $g(z) = g_\alpha(z)$ for $z \in Z_\alpha$
  • Verifies that $(Z, g)$ is indeed an upper bound for the chain
• Applies Zorn's Lemma to conclude that $\mathcal{F}$ has a maximal element $(Z_0, g_0)$
• Argues that $Z_0$ must be equal to the entire space $X$, completing the proof of the theorem

Significance in functional extension

• Enables the extension of bounded linear functionals from a subspace to the entire normed linear space
• Ensures that the extended functional preserves the norm of the original functional
• Provides a powerful tool for studying dual spaces and weak topologies in functional analysis
• Guarantees the existence of the extension without explicitly constructing it

Applications to normed spaces

• Applies to the space $C[0,1]$ of continuous functions on the interval $[0,1]$ with the supremum norm
  • Considers the subspace $Y$ of polynomials and defines $f(p) = p(0)$ for polynomials $p \in Y$
  • Asserts the existence of a bounded linear functional $F$ on $C[0,1]$ that extends $f$
• Extends to the space $\ell^1$ of absolutely summable sequences with the $\ell^1$ norm
  • Takes the subspace $Y$ of sequences with only finitely many non-zero terms
  • Defines a functional $f((x_n)) = \sum_{n=1}^\infty a_n x_n$ for sequences $(x_n) \in Y$, where $(a_n)$ is a sequence in $\ell^\infty$
  • Confirms the existence of an extension of $f$ to a bounded linear functional on the entire space $\ell^1$