This course develops the theory of Banach and Hilbert spaces, extending finite-dimensional linear algebra to infinite dimensions. Beginning with the Hahn-Banach extension theorem—a cornerstone result asserting the richness of the dual space—we establish the fundamental duality principle that underlies all of functional analysis. We then concretely study the duals of classical function spaces like $L^p$ and $C(K)$, before ascending to the more abstract study of weak topologies and the geometry of convex sets, culminating in the Krein-Milman theorem. This foundation naturally leads us to Banach algebras, which provide an algebraic framework unifying both operators and functions under a single structure.

The latter chapters reveal the power of this framework through functional calculus and spectral theory. We develop the holomorphic functional calculus for elements of a Banach algebra, showing how to evaluate analytic functions of operators, and then specialize to $C^*$-algebras—a class rich enough to capture quantum mechanics and harmonic analysis. The Borel functional calculus extends these ideas to normal operators, and the spectral theorem emerges as a unifying principle: a normal operator on a Hilbert space can be understood as multiplication by a function on a measure space. Throughout, duality and weak topologies serve as connecting threads, while spectral theory reveals the hidden diagonal structure of infinite-dimensional operators.

# 1. Hahn-Banach Extension Theorems

This chapter develops the Hahn-Banach extension theorems and their major consequences. The central question is deceptively simple: given a bounded linear functional defined on a subspace, can it be extended to the whole space without increasing its norm? The answer — yes, always — has far-reaching structural consequences for the geometry of Banach spaces. We also introduce the bidual, reflexivity, dual operators, quotient spaces, locally convex spaces, and several landmark corollaries that reveal how richly populated every dual space must be.

## Dual Spaces and Basic Notation

The first thing to establish is how large the dual of a normed space really is. Could it happen that $X^*$ is trivial, containing only the zero functional? This would be catastrophic: we could not distinguish any two points of $X$ by linear measurements, and nearly all of functional analysis would collapse. The Hahn-Banach theorem, proved below, guarantees that $X^*$ is always abundantly populated. But before proving that, we fix notation.

Let $X$ be a normed space (real or complex). The **dual space** of $X$ is

\begin{align*} X^* = \{ f : X \to \mathbb{R} \text{ or } \mathbb{C} : f \text{ is linear and continuous} \}. \end{align*}

Since $f$ is linear, continuity is equivalent to boundedness, so we may equivalently define $X^*$ as the space of bounded linear functionals on $X$. The dual is always a Banach space under the norm

\begin{align*} \|f\|_{X^*} := \sup\{|f(x)| : x \in B_X\}, \end{align*}

where $B_X := \{x \in X : \|x\|_X \le 1\}$ is the closed unit ball and $S_X := \{x \in X : \|x\|_X = 1\}$ is the unit sphere.

[example: Standard Dual Space Identifications] The following classical identifications describe the duals of familiar sequence spaces. (i) $\ell_p^* \cong \ell_q$ for $1 \le p < \infty$, where $q$ is the Hölder conjugate of $p$, i.e. $\frac{1}{p} + \frac{1}{q} = 1$. The isomorphism sends $(y_n) \in \ell_q$ to the functional $f_y(x) = \sum_n x_n y_n$. To see this is isometric, the key estimate is Hölder's inequality: $|f_y(x)| = |\sum_n x_n y_n| \le \|x\|_{\ell_p} \|y\|_{\ell_q}$, so $\|f_y\|_{\ell_p^*} \le \|y\|_{\ell_q}$. Equality is achieved by the sequence $x_n^* = |y_n|^{q-2}\bar{y}_n/\|y\|_{\ell_q}^{q-1}$ (when $y \ne 0$), which satisfies $\|x^*\|_{\ell_p} = 1$ and $f_y(x^*) = \|y\|_{\ell_q}$, establishing $\|f_y\|_{\ell_p^*} = \|y\|_{\ell_q}$. (ii) $c_0^* \cong \ell_1$. Here the natural pairing is again $f_y(x) = \sum_n x_n y_n$ for $y = (y_n) \in \ell_1$ and $x = (x_n) \in c_0$. To verify surjectivity: given any $f \in c_0^*$, let $e_n$ be the standard basis vectors in $c_0$ and set $y_n = f(e_n)$. Then for any finite sum $x = \sum_{n=1}^N a_n e_n$ we have $f(x) = \sum_n a_n y_n$, and the isometry $\|f\| = \|y\|_{\ell_1}$ follows because we can choose signs to make $|\sum_n y_n a_n| = \sum_n |y_n|$ for any truncation. (iii) If $H$ is a Hilbert space, then $H^* \cong H$ (conjugate-linearly in the complex case), by the Riesz representation theorem. Every $f \in H^*$ is uniquely of the form $f = (\cdot, u)_H$ for some $u \in H$, with $\|f\| = \|u\|_H$. The identification (iii) underpins the pairing notation introduced below. [/example]

**Notation for isomorphism and pairing.** For normed spaces $X$ and $Y$:

- $X \sim Y$ means $X$ and $Y$ are isomorphic, i.e. there exists a linear homeomorphism $T: X \to Y$. - $X \cong Y$ means $X$ and $Y$ are isometrically isomorphic, i.e. there exists a surjective linear $T: X \to Y$ with $\|T(x)\|_Y = \|x\|_X$ for all $x \in X$ (which forces $T$ to be injective, hence bijective, with continuous inverse). - For $x \in X$ and $f \in X^*$, we write the **duality pairing**

\begin{align*} f(x) \end{align*}

using ordinary function notation. (Some sources use angle brackets $\langle x, f \rangle$; per our notation standards, we write $f(x)$ instead.)

The open mapping theorem gives a useful shortcut: if $X$ and $Y$ are both Banach, then to show $T: X \to Y$ is an isomorphism it suffices to check that one of $T$ or $T^{-1}$ is continuous, because the other continuity is then automatic.

By the definition of $\|f\|_{X^*}$, we always have the fundamental estimate

\begin{align*} |f(x)| \le \|f\|_{X^*} \|x\|_X. \end{align*}

## The Hahn-Banach Theorem