### Vector integration

Lately I've been thinking a lot about algebras of operators acting on a Hilbert space, since they provide an extremely useful tool for thinking about locality in quantum field theory. I'm working on a review article about Tomita-Takesaki modular theory to supplement my recent review on the type classification of von Neumann algebras. The core object of study in Tomita-Takesaki theory is a one-parameter group of unitary operators $\Delta^{it},$ generated by a single positive (often unbounded) operator $\Delta.$

In physics, the Tomita-Takesaki unitaries furnish a "hidden thermodynamic symmetry" of a physical state. A lot of interesting physics and mathematics can be learned by studying the analytic structure of the function $z \mapsto \Delta^{z}$ for generic complex $z,$ or of the function $z \mapsto \Delta^{z} |\psi\rangle$ for generic complex $z$ and some fixed state $|\psi\rangle.$ But in order to do this, we need to understand how to do complex analysis for operator-valued (or vector-valued) functions.

In ordinary complex analysis as we learn it in undergrad, a fundamental role is played by Cauchy's two integral theorems, which say that when $f$ is a holomorphic function in a region bounded by a curve $\gamma,$ we have
$$0 = \int_{\gamma} d\zeta\, f(\zeta)$$
and
$$f(z) = \frac{1}{2 \pi i} \int_{\gamma} d\zeta\, \frac{f(\zeta)}{z - \zeta}.$$
Most of the important theorems in complex analysis are proved by rewriting expressions in terms of contour integrals and then manipulating those integrals. To develop a theory of complex analysis for vector- or operator-valued functions, it is therefore essential to begin with a theory of integration for those functions.

This post is about the Bochner integral, which is a way of defining integration for functions valued in a Banach space (which could be, for example, a Hilbert space or the space of bounded operators acting on that Hilbert space). The mathematical details of constructing the Bochner integral are interesting, but not by themselves particularly important for physics. What is important is understanding: "What manipulations am I allowed to do with a Banach space integral?" The purpose of this post is to outline those rules, and to explain some of their consequences for complex analysis of Banach-valued functions. In a companion post, I will use these ideas to explain Stone's theorem in the language of complex analysis. Stone's theorem tells us that every one-parameter group of unitaries on Hilbert space, satisfying an appropriate continuity condition, can be written in the form $t \mapsto \Delta^{it}$ for some operator $\Delta$; it also tells us about the structure of the analytic continuation $\Delta^{z}.$

The outline is:

1. In section 1, I will briefly review the essential features of Lebesgue integration; much more detail can be found in my post on $L^p$ spaces.
2. In section 2, I will define the class of "Bochner measurable" Banach-valued functions on which integration can in-principle be defined, and give a simple criterion for checking whether a Bochner-measurable function has finite integral. I will also show that the Bochner integral is linear and subadditive, and show that continuous functions are always Bochner measurable.
3. In section 3, I will describe a few important manipulations that can be done with Bochner integrals. In particular, I will explain when a linear operator acting on an integral can be moved inside the integrand, and when the order of an integral and a sum can be interchanged.
4. In section 4, I will prove some basic theorems of complex analysis for Banach-valued functions.

I learned about the Bochner integral from Wikipedia, Yosida, and Dunford and Schwartz. I learned about Lebesgue integration, which is a prerequisite for the Bochner theory, from Rudin.

Prerequisites: Definition of a Banach space; definition of operator norm. Single-variable complex analysis. Comfort with abstract integration and measure spaces.

## 1. Lightning review of Lebesgue integration

Let $\Omega$ be a generic set. To define integration for functions with base space $\Omega$, we need to assign a notion of measure to subsets of $\Omega.$ This means we need to endow $\Omega$ with the structure of a measure space. Concretely, we specify a $\sigma$-algebra $\Sigma,$ which is a collection of subsets of $\Omega$ that we have declared to be "measurable." We also specify a measure $\mu,$ which is a function $\mu : \Sigma \to [0, \infty]$ that satisfies appropriate additivity criteria. Given a topological space $X$, a function $f : \Omega \to X$ is said to be measurable if $f^{-1}(O)$ is measurable for every open set $O \subseteq X.$

Given a measure space $\Omega$, we can try to define the integral of a measurable function $f : \Omega \to \mathbb{C}.$ The theory of integration for generic measurable functions is built up out of a theory of integration for simple functions, for which the definition of the integral is obvious. A function $s : \Omega \to \mathbb{C}$ is said to be simple if its range contains only finitely many elements $\{\alpha_1, \dots, \alpha_n\},$ and if for every nonzero $\alpha_j,$ the set $s^{-1}(\alpha_j)$ has finite measure. The integral of $s$ is then naturally defined as
$$\int d\mu\, s = \sum_{j} \alpha_j \cdot \mu(s^{-1}(\alpha_j)).$$
To define integration for a generic measurable function, one goes through the following steps:
1. One shows that every measurable function $f : \Omega \to \mathbb{C}$ can be written as the pointwise limit of a sequence of simple functions $s_n$ with $|s_1| \leq |s_2| \leq \dots \leq |f|.$
2. One defines the integral of a measurable function $f : \Omega \to [0, \infty]$ as the limit of $\int d\mu\, s_n$ for a sequence of positive simple functions that approximate $f$ from below; because $\int d\mu\, s_n$ is a monotonically increasing positive sequence, the limit exists (though it may be infinity). One shows that this limit is independent of the sequence of simple functions used to approximate $f$.
3. One shows that for any measurable function $f : \Omega \to \mathbb{C},$ if $\int d\mu\, |f|$ is finite, then $\int d\mu\, f$ can be defined uniquely as the linear combination of the integrals of the positive/negative parts of its real/imaginary parts. One also shows that $\int d\mu\, f$ is the limit of $\int d\mu\, s_n$ for a sequence of simple functions approximating $f.$
4. One proves various other theorems about the Lebesgue integral: $\int d\mu\, (f + g) = \int d\mu\, f + \int d\mu\, g$ and $|\int d\mu\, f| \leq \int d\mu\, |f|$.

## 2. The Bochner integral

A Banach space $X$ is a vector space with a norm $\lVert \cdot \rVert,$ such that $X$ is complete in the topology induced by the norm. Some important examples of Banach spaces are:
1. Any Hilbert space $\mathcal{H}$ with the norm $$\lVert |\psi \rangle \rVert = \sqrt{\langle \psi | \psi \rangle}.$$
2. The set of bounded operators on any Hilbert space, $\mathcal{B}(\mathcal{H}),$ with norm
$$\lVert O \rVert_{\infty} = \sup_{\lVert |\psi\rangle \rVert \leq 1} \lVert O |\psi\rangle \rVert.$$
3. The complex numbers $\mathbb{C}$ with norm given by the absolute value.
Given a measure space $\Omega$ and a Banach space $X$, we would like to understand when a function $f : \Omega \to X$ can be assigned a consistent integral $\int d\mu\, f.$ As in the case of Lebesgue integration, we will start with simple functions.

A function $s : \Omega \to X$ is simple if its range $\{x_1, \dots, x_n\}$ is finite, and $s^{-1}(x_j)$ has finite measure for any $x_j \neq 0.$ The integral of a simple function is defined as
$$\int d\mu\, s = \sum_{x \in \text{range}(s) - \{0\}} x \cdot \mu(s^{-1}(x)).$$
It is then a straightforward exercise (using basic properties of the measure $\mu$) to show that the integration of simple functions satisfies the important properties of linearity
$$\int d\mu\, (s + r) = \int d\mu\, s + \int d\mu\, r$$
$$\lVert \int d\mu\, s \rVert \leq \int d\mu\, \lVert s \rVert.$$
Note that on the right-hand side of this last expression, the integral is an ordinary Lebesgue integral.

Now that we have a theory of integration for simple functions, we want to use it to define integration for more general functions. There is a subtlety that arises in the case of a generic Banach space $X$ that was not present for $\mathbb{C},$ which is that not every measurable function $f : \Omega \to X$ can be approximated by simple functions. This leads us to define a new notion of measurability: a function $f : \Omega \to X$ is said to be Bochner measurable if there exists a sequence of simple functions $s_n : \Omega \to X$ that converges pointwise to $f$ almost everywhere. If there exists such a sequence with the additional property $\int d\mu\, \lVert f - s_n \rVert \to 0,$ then we say $f$ is Bochner integrable, and we define $\int d\mu\, f = \lim_n \int d\mu\, s_n.$

There are a few things we need to verify in order to see that this definition of the integral makes sense. I'll list them below.
• We need to verify that the function $\lVert f - s_n \rVert$ is measurable, so that it makes sense to ask whether $\int d\mu\, \lVert f - s_n \rVert$ converges to zero.
• We need to verify that when $f$ is Bochner integrable, the limit $\int d\mu\, s_n$ exists.
• We need to verify that this limit is independent of the defining sequence $s_n.$
• We need to verify that this definition of the integral has the crucial properties of linearity and subadditivity.
• We don't need to do this, but it would be nice to have a simpler criterion for integrability than "construct a sequence of simple functions with some special properties."
I won't give all of the details of each of these bullet points, but I will sketch the important pieces of the proofs.

One can show that the functions $\lVert f \rVert$ and $\lVert f - s_n \rVert$ are measurable by using the fact that the pointwise limit of a sequence of measurable functions is measurable.

When $f$ is Bochner integrable, one can show that $\lim_n \int d\mu\, s_n$ exists by showing that $\int d\mu\, s_n$ is a Cauchy sequence. This follows from linearity and subadditivity of Bochner integration of simple functions, and of Lebesgue integration of measurable functions:
$$\lVert \int d\mu\, s_m - \int d\mu\, s_n \rVert \leq \int d\mu\, \lVert s_m - s_n \rVert \leq \int d\mu\, \lVert s_m - f \rVert + \int d\mu\, \lVert s_n - f \rVert \to 0.$$

A similar argument shows that $\int d\mu\, f$ is independent of the defining sequence of simple functions. If $s_n$ and $r_n$ are sequences of simple functions that converge pointwise to $f$ almost everywhere, and that satisfy $\int d\mu\, \lVert f - s_n \rVert \to 0,$ $\int d\mu\, \lVert f - r_n \rVert \to 0,$ then we have
$$\lVert \int d\mu\, r_n - \int d\mu\, s_n \rVert \leq \int d\mu\, \lVert r_n - s_n \rVert \leq \int d\mu\, \lVert r_n - f \rVert + \int d\mu\, \lVert s_n - f \rVert \to 0.$$

Linearity and subadditivity of the Bochner integral are easy to show by taking limits of the analogous properties for simple functions.

Finally, we would like to have a good criterion for when a function is Bochner integrable that doesn't require explicitly constructing a sequence of defining functions. I will do this in two steps:
• First, I will show that if $\Omega$ is a Borel subset of $\mathbb{R}^m$ and $\mu$ is a Borel measure, then any continuous function $f : \Omega \to X$ is Bochner measurable.
• Second, I will show that a Bochner measurable function for any measure space $f : \Omega \to X$ is Bochner integrable iff we have $\int d\mu\, \lVert f \rVert < \infty.$
The first step is pretty easy; given a continuous function $f : \Omega \to \mathbb{R}^m$, we just need to construct a sequence of simple functions that converges pointwise almost everywhere to $\mathbb{R}^m.$ What we will do is, for each integer $n,$ consider the cubic lattice on $\mathbb{R}^m$ of spacing $1/n,$ and the ball $B_n$ of radius $n.$ We construct a simple function $s_n$ by assigning, to each cube in the lattice that intersects both $B_n$ and $\Omega,$ the value $f(\xi)$ for some point $\xi$ in the intersection of the cube, $B_n,$ and $\Omega.$ Since only finitely many cubes intersect $B_n,$ the function $s_n$ is simple. It is easy to show using continuity of $f$ that the sequence $s_n$ converges pointwise to $f.$

The second part is a little more subtle. First, if $f : \Omega \to X$ is Bochner integrable, and $s_n$ is a defining sequence of simple functions, then we have
$$\int d\mu\, \lVert f \rVert \leq \int d\mu\, \lVert f - s_n \rVert + \int d\mu\, \lVert s_n \rVert,$$
and this must be finite for large enough $n,$ since $\int d\mu\, \lVert f - s_n \rVert$ goes to zero. So every Bochner integrable function satisfies $\int d\mu\, \lVert f \rVert < \infty.$ Conversely, if $f$ is Bochner integrable and $s_n$ is a sequence of simple functions that converges pointwise to $f$ almost everywhere, and we have $\int d\mu\, \lVert f \rVert < \infty,$ then we want to show that $f$ is Bochner integrable. This requires constructing another sequence of simple functions $r_n,$ which converges pointwise to $f$ almost everywhere and satisfies $\int d\mu\, \lVert f - r_n \rVert \to 0.$

To construct this sequence, we basically want to set $r_n$ equal to $s_n$ when $s_n$ is very close to $f,$ and zero when $s_n$ is not close to $f.$ Fix some $\epsilon$ with $0 < \epsilon < 1,$ and define
$$r_n(x) = \begin{cases} s_n(x) & \lVert s_n(x) \rVert \leq (1 + \epsilon) \lVert f(x) \rVert \\ 0 & \text{else} \end{cases}.$$
It is easy to verify that $s_n$ is simple and that it converges pointwise to $f$ wherever $s_n$ does. Finally, we have $\lVert r_n - f \rVert \leq (2 + \epsilon) \lVert f \rVert.$ So by integrability of $\lVert f \rVert$ and the dominated convergence theorem, we have $\int d\mu\, \lVert r_n - f \rVert \to \int d\mu\, \lVert f - f \rVert = 0.$

## 3. Basic manipulations

Here are the manipulations I would like to show are legal with Bochner integrals:
• If $f : \Omega \to X$ is Bochner integrable and $T : X \to Y$ is linear and bounded, then $T f$ is Bochner integrable and we have $T \int d\mu\, f = \int d\mu\, T f.$
• If $f : \Omega \to X$ is Bochner integrable, $T : X \to Y$ is unbounded but closed (I'll define this later), the image of $f$ lies in the domain of $T$, and $T f$ is Bochner integrable, then we have $T \int d\mu\, f = \int d\mu\, T f.$
• If $\Omega$ is a finite measure space, $f : \Omega \to X$ is Bochner integrable, and $f$ can be written almost everywhere as a series $f(\omega) = \sum_{n} a_n(\omega)$ of Bochner integrable functions such that the series converges absolutely and uniformly, then we have $\int d\mu\, f = \sum_{n} \int d\mu\, a_n.$ (This is a special case of the more general Fubini theorem, but the general case has a less inuitive proof, and we only need the special case for complex analysis.)
For the first bullet point, let $X$ and $Y$ be Banach spaces, and let $T$ be linear and bounded. If $s_n$ is a defining sequence for the Bochner integrable function $f$ (meaning it converges pointwise almost everywhere to $f$ and satisfies $\int d\mu\, \lVert s_n - f \rVert \to 0$), then it is easy to check by boundedness of $T$ that $T s_n$ is a defining sequence for $T f.$ Linearity of $T$ then easily gives
$$T \int d\mu\, f = \lim_n T \int d\mu\, s_n = \lim_n \int d\mu\, T s_n = \int d\mu\, T f.$$

The second bullet point is more subtle, but very important. First, let me be more precise about what a closed operator between Banach spaces is. It is often very useful to consider linear operators from $X$ to $Y$ that are not defined on all of $X$; they are defined instead on some linear subspace $D_T \subseteq X,$ which is often taken to be dense within $X$. These operators are generally not taken to be bounded on $D_T$; if they were, then they could be extended to all of $X$ by continuity. A closed operator is an unbounded operator $T : D_T \to Y$ that has one of the essential properties of bounded operators. If $T$ were bounded, then whenever we have $x_n \to x,$ we have $T x_n \to T x.$ A closed operator has the weaker property that if $x_n \to x$ is in $D_T$ and $T x_n$ converges to something, then $T x_n$ must converge to $T x.$ So closed operators need not take all convergent sequences to convergent sequences, but when they do, the image sequence and converges to the image of the limit of the domain sequence. Closed operators are very important in operator theory; for example, every self-adjoint operator on Hilbert space is closed.

The second bullet point above is known as Hille's theorem. Let us state it again. We will assume that $T : D_T \to Y$ is a closed operator, that $f : \Omega \to X$ is a Bochner integrable function, that the image of $f$ lies in $D_T,$ and that $T f$ is Bochner integrable. We then want to show the identity $T \int d\mu\, f = \int d\mu\, T f.$ The key will be to go to the direct sum space $X \oplus Y.$ Any book on Banach space theory will tell you that $X \oplusY$ is a Banach space with respect to the norm
$$\lVert x \oplus y \rVert = \lVert x \rVert \oplus \lVert y \rVert.$$
It is easy to check that the operator $T$ is closed if and only if the set
$$\Gamma_T = \{x \oplus T x | x \in D_T\} \subseteq X \oplus Y$$
is closed in the Banach space topology. Since $T$ is a closed operator, the set $\Gamma_T$ is a closed linear subspace of $X \oplus Y,$ so it is itself a Banach space. Consider the function $g : \Omega \to \Gamma_T$ defined by
$$g(\omega) = f(\omega) \oplus T f(\omega).$$
Since $f$ and $T f$ are both Bochner integrable, there exist sequences $r_n$ and $s_n$ such that $\int d\mu\, \lVert f - r_n \rVert \to 0$ and $\int d\mu\, \lVert T f - s_n \rVert \to 0.$
It is then easy to show that we have
$$\int d\mu\, \lVert r_n \oplus s_n - g \rVert \to 0.$$
So $g$ is Bochner integrable, and we have
$$\int d\mu\, g = \lim_n \int d\mu\, r_n \oplus \lim_n \int d\mu\, s_n.$$
Since each $\int d\mu\, r_n \oplus \int d\mu\, s_n$ is in $\Gamma_T,$ and $\Gamma_T$ is closed, the limit is also in $\Gamma_T.$ So there exists some $x_0 \oplus T x_0$ in $\Gamma_T$ with
$$\int d\mu\, f \oplus \int d\mu\, T f = \int d\mu\, g = x_0 \oplus T x_0.$$
By matching the left and right side, we easily conclude $T \int d\mu\, f = \int d\mu\, T f.$

Finally, it is convenient to have a rule for when we can interchange a sum and an integral. Suppose that $\Omega$ is a finite measure space and $f : \Omega \to X$ is a Bochner integrable function. Suppose further that it can be written almost everywhere as a series of Bochner integrable functions, $f = \sum_n a_n,$ such that the series converges absolutely uniformly. I.e., for any $\epsilon,$ there exists some $N$ such that we have
$$\sup_{z \in \Omega} \sum_{n=N}^{\infty} \lVert a_n \rVert < \epsilon.$$
We will show that we have
$$\sum_n \int d\mu\, a_n = \lim_{N} \int d\mu\, S_N = \int d\mu\, \lim_N S_N = \int d\mu\, f.$$

To see this, fix some integer $M$, and observe:
$$\lVert \int d\mu\, \lim_N S_N - \int d\mu\, S_M \rVert = \lVert \int d\mu\, \sum_{n=M+1}^{\infty} a_n \rVert \leq \int d\mu\, \sum_{n=M+1}^{\infty} \lVert a_n \rVert.$$
The assumption of uniform absolute convergence, together with finiteness of the measure $\mu,$ implies that this goes to zero for $M$ large.

## 4. Complex analysis for Banach-valued functions

Now, let $\Omega \subseteq \mathbb{C}$ be some open subset of the complex numbers, and let $X$ be a Banach space. A function $f : \Omega \to X$ is said to be holomorphic at the point $z \in \Omega$ if the limit
$$f'(z) = \lim_{h \to 0} \frac{f(z+h) - f(z)}{h}$$
exists.
Furthermore, given any continuous function $f : \Omega \to X$ and a simple oriented curve $t \mapsto \gamma(t)$ in $\Omega,$ we can define the contour integral
$$\int_{\gamma} dz\, f(z) \equiv \int_{0}^{1} dt\, \gamma'(t) f(\gamma(t)).$$
Differentiability of $\gamma$ and continuity of $f$ guarantee that $t \mapsto \gamma'(t) f(\gamma(t))$ is Bochner measurable, and compactness of $[0, 1]$ further guarantees that the integral $\int_{0}^{1} dt\, |\gamma'(t)| \lVert f(\gamma(t)) \rVert$ is finite, so the contour integral is well defined as a Bochner integral. Usual checks show that it is independent of the parametrization of the curve $\gamma.$
We can now show that all of the usual relationships between contour integrals and holomorphic functions hold for Banach-valued functions. In particular, we will show:
• If $f : \Omega \to X$ is holomorphic, $\Omega$ is simply connected, and $\gamma$ is a simple closed curve in $\Omega,$ then we have $\int_{\gamma} dz\, f = 0.$
• If $f : \Omega \to X$ is holomorphic and $\Omega$ is simply connected, then for any point $z$ in $\Omega$ and any simple curve $\gamma$ surrounding $z$ with clockwise orientation, we have
$$f(z) = \frac{1}{2 \pi i} \int_{\gamma} dw\, \frac{f(w)}{w - z}.$$
• If $f : \Omega \to X$ is holomorphic, then it is analytic and hence infinitely differentiable.
• If $f : \Omega \to X$ is continuous and $\int_{\gamma} dz\, f(z)$ vanishes for every simple closed curve $\gamma$ in $f,$ then $f$ is holomorphic.
• If $f_n : \bar{\Omega} \to X$ is a sequence of holomorphic functions continuous on the boundary of $\bar{\Omega}$ that converge uniformly to $f : \bar{\Omega} \to X,$ then $f$ is holomorphic in $\Omega$ and continuous on its boundary.
The first two theorems are simple consequences of the weak version of Hille's theorem discussed in the previous subsection. For any bounded linear functional $\Lambda : f \to X,$ we have
$$\Lambda\left( \int_{\gamma} dz\, f \right) = \int_{\gamma} dz\, \Lambda f = 0,$$
since it is easy to show using linearity and boundedness of $\Lambda$ that $\Lambda f$ is holomorphic in the usual sense. One then appeals to the standard fact in Banach spaces that if $\Lambda(x)$ vanishes for every bounded linear functional $\Lambda,$ we must have $x = 0.$ Similarly, we have
$$\Lambda\left( \frac{1}{2 \pi i} \int_{\gamma} dw\, \frac{f(w)}{w-z} \right) = \frac{1}{2 \pi i} \int_{\gamma} dw\, \frac{\Lambda f(w)}{w-z} = \Lambda(f(z)),$$
hence
$$\Lambda\left( \frac{1}{2 \pi i} \int_{\gamma} dw\, \frac{f(w)}{w-z} \right) = f(z).$$

The third theorem can be proven using the same argument used to prove the equivalence of holomorphic and analytic functions in ordinary complex analysis, together with our knowledge from the previous section of when integrals and sums can be interchanged. Let $f$ be holomorphic at $z_0,$ and consider a circle $C$ of radius $r$ centered at $z_0$ such that the whole closed disc it contains lies in $\Omega.$ Then for any $z$ in that disc, we have
$$f(z) = \frac{1}{2\pi i} \int_{C} dw\, \frac{f(w)}{w - z} = \frac{1}{2 \pi i} \int_C dw\, \frac{1}{w-z_0} \frac{f(w)}{1 - \frac{z-z_0}{w - z_0}}.$$
Expanding the integrand in terms of a geometric series, we have
$$f(z) = \frac{1}{2 \pi i} \int_C dw\, \sum_n \frac{f(w)}{w-z_0} \left(\frac{z-z_0}{w-w_0}\right)^n.$$
From the considerations of the previous section, we can interchange the sum and the integral if we know that the series $\sum_n \left( \frac{z - z_0}{w - w_0} \right)^n$ converges absolutely uniformly on the contour for $w.$ This is easy to show using the usual basic manipulations of geometric series.
So we have
$$f(z) = \frac{1}{2 \pi i} \sum_n int_C dw\, \frac{f(w)}{w-z_0} \left(\frac{z-z_0}{w-w_0}\right)^n.$$
Which gives an explicit expression for $f$ in terms of a power series, and therefore shows that $f(z)$ is analytic.

For the fourth theorem, we assume without loss of generality that $\Omega$ is connected. We pick a point $z_0 \in \Omega,$ and define $F(z) = \int_{\gamma} dw\, f(w)$ for a curve $\gamma$ connecting $z_0$ to $z,$ where by assumption the function $F(z)$ is independent of the choice of curve $\gamma.$ For any small complex number $h,$ the difference $F(z+h) - F(z)$ is equal to the integral of $f$ along the straight line connecting $z$ to $z+h.$ From this it is easy to see that we have
$$\frac{F(z+h) - F(z)}{h} = \int_0^1 dt\, f(z+th).$$
It is easy to see from continuity of $f$ that this converges to $f(z).$ This tells us that $F(z)$ is holomorphic with derivative $f(z)$; since we have already shown that holomorphic functions are analytic and hence infinitely differentiable, this implies that $f(z)$ is holomorhpic.

Finally, let $\Omega$ be an open set in $\mathbb{C}$, and let $f_n : \bar{\Omega} \to X$ be a sequence of functions that are holomorphic on $\Omega$ and continuous on $\bar{\Omega}.$ Furthermore suppose that $f_n$ converges uniformly to $f$ on $\bar{\Omega}.$ We want to show that $f$ is holomorphic on $\Omega$ and continuous on $\bar{\Omega}.$

To see continuity, fix any $z_0 \in \bar{\Omega}$ and any $\epsilon > 0.$ We want to show that there exists some $\delta > 0$ such that for every $z \in \bar{\Omega}$ with $\lVert z - z_0 \rVert < \delta,$ we have $\lVert f(z) - f(z_0) \rVert < \epsilon.$ But we have
$$\lVert f(z) - f(z_0) \rVert \leq \lVert f(z) - f_n(z) \rVert + \lVert f_n(z) - f_n(z_0) \rVert + \lVert f_n(z_0) - f(z_0) \rVert.$$
By the assumption of uniform convergence, the first and third terms can be made arbitrarily small and independent of $z, z_0$ by taking $n$ to be large. After taking some fixed large value of $n,$ continuity of the function $f_n$ can be used to make the middle term arbitrarily small by choosing $z$ close to $z_0.$

To see holomorphy, we note that to show that $f$ is holomorphic at a point $z \in \Omega,$ it suffices to show that it is holomorphic in a small disc containing $z$ and contained in $\Omega.$ But within this simply connected disc, holomorphy of the functions $f_n(z)$ implies $\int_{\gamma} dz\, f_n = 0$ for any simple curve $\gamma.$ We then note that for any such curve, we have
$$\lVert \int_{\gamma} dz\, f \rVert = \lVert \int_{\gamma} dz\, (f - f_n) \rVert \leq \int_{\gamma} dz\, \lVert f - f_n \rVert.$$
The right hand side goes to zero due to uniform convergence of $f_n$ to $f$, so we have $\int_{\gamma} dz\, f = 0$ for every $\gamma$ in the disc, and so $f$ is holomorphic by the previous theorem.

Complex analysis, as we usually learn it, is the study of differentiable functions from $\mathbb{C}$ to $\mathbb{C}$. These functions have many nice properties: if they are differentiable even once then they are infinitely differentiable; in fact they are analytic, meaning they can be represented in the vicinity of any point as an absolutely convergent power series; moreover at any point $z_0$, the power series has radius of convergence equal to the radius of the biggest disc centered at $z_0$ which can be embedded in the domain of the function. The same basic properties hold for differentiable functions in higher complex dimensions. If $\Omega$ is a domain --- i.e., a connected open set --- in $\mathbb{C}^n$, and $f : \Omega \to \mathbb{C}^n$ is once differentiable, then it is in fact analytic, and can be represented as a power series in a neighborhood of any point $z_*$, i.e., we have an expression like $$f(z) = \sum a_{k_1 \dots k_n} (z_1 - z_*)^{k_1} \dots (z_n - z_*)^{k_n}.$$ The
Stone's theorem is the basic result describing group-like unitary flows on Hilbert space. If the map $t \mapsto U(t)$ is continuous in a sense we will make precise later, and each $U(t)$ is a unitary map on a Hilbert space $\mathcal{H},$ and we have $U(t+s)=U(t)U(s),$ then Stone's theorem asserts the existence of a (self-adjoint, positive definite, unbounded) operator $\Delta$ satisfying $U(t) = \Delta^{it}.$ This reduces the study of group-like unitary flows to the study of (self-adjoint, etc etc) operators. Quantum mechanically, it tells us that every group-like unitary evolution is generated by a time-independent Hamiltonian. This lets us study very general symmetry transformations in terms of Hamiltonians. The standard proof of Stone's theorem, which you'll see if you look at Wikipedia , involves trying to make sense of a limit like $\lim_{t \to 0} (U(t) - 1)/t$. However, I have recently learned of a beautiful proof of Stone's theorem that works instead by stud