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Definition 1.1.1. Let $G$ be an open subset of the complex plane. A continuous real-valued function $u: G \rightarrow \mathbb{R}$ satisfies the mean value property if, whenever $\overline{B(a, r)}={z \in \mathbb{C}:|z-a| \leq r}$ is contained in $G$, then
$$u(a)=\frac{1}{2 \pi} \int_0^{2 \pi} u\left(a+r e^{i \theta}\right) d \theta$$

Definition 1.1.1. Let $G$ be an open subset of the complex plane. A continuous real-valued function $u: G \rightarrow \mathbb{R}$ satisfies the mean value property if, whenever $\overline{B(a, r)}={z \in \mathbb{C}:|z-a| \leq r}$ is contained in $G$, then
$$u(a)=\frac{1}{2 \pi} \int_0^{2 \pi} u\left(a+r e^{i \theta}\right) d \theta.$$

In this definition, $u(a)$ represents the value of the function $u$ at the point $a$, and $u\left(a+re^{i\theta}\right)$ represents the value of $u$ at points on the circle centered at $a$ with radius $r$, parameterized by $\theta$ from $0$ to $2\pi$. The mean value property states that the value of $u$ at the center $a$ is equal to the average of the values of $u$ on the circle $|z-a|=r$. This holds for any $r>0$ as long as the closed ball $\overline{B(a,r)}$ is contained within the open set $G$.

Definition 1.2.1. For a compact Hausdorff space $X$, let $C(X)$ denote the space of continuous complex-valued functions on $X$ with the supremum norm. A function algebra on $X$ is a closed subalgebra of $C(X)$ that contains the constant functions and separates the points of $X$.

Definition 1.2.1. For a compact Hausdorff space $X$, let $C(X)$ denote the space of continuous complex-valued functions on $X$ with the supremum norm. A function algebra on $X$ is a closed subalgebra of $C(X)$ that contains the constant functions and separates the points of $X$.

In this definition, $C(X)$ represents the space of continuous functions from $X$ to the complex numbers. The supremum norm is a norm on $C(X)$ defined as the maximum absolute value of the function over $X$.

A function algebra on $X$ is a subset of $C(X)$ that is closed under addition, scalar multiplication, and function multiplication. It contains all constant functions, meaning it includes functions of the form $f(x) = c$ for any constant $c$. Additionally, a function algebra separates the points of $X$, which means that for any distinct points $x_1$ and $x_2$ in $X$, there exists a function $f$ in the algebra such that $f(x_1) \neq f(x_2)$.

Furthermore, a function algebra is required to be closed, meaning that it contains all limits of sequences or nets of functions in the algebra. This ensures that the algebra remains a closed subset of $C(X)$.

Overall, a function algebra on $X$ is a closed subalgebra of $C(X)$ that includes constant functions and separates the points of $X$, providing a rich collection of functions for studying the properties of the compact Hausdorff space $X$.

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