# 数学代写|偏微分方程代写partial difference equations代考|Bounded linear functionals on $L^p(X)$ and weak convergence

#### Doug I. Jones

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## 数学代写|偏微分方程代写partial difference equations代考|§8 Bounded linear functionals on $L^p(X)$ and weak convergence

We begin with
Theorem 1. (Extension of linear functionals)
Take $p \in[1,+\infty)$ and let $A: M^{\infty}(X) \rightarrow \mathbb{R}$ denote a linear functional with the following property: We have a constant $\alpha \in[0,+\infty)$ such that
$$|A(f)| \leq \alpha|f|_{L^p(X)} \quad \text { for all } \quad f \in M^{\infty}(X)$$
holds true. Then there exists exactly one bounded linear functional $\widehat{A}$ : $L^p(X) \rightarrow \mathbb{R}$ satisfying
$$|\widehat{A}| \leq \alpha \quad \text { and } \quad \widehat{A}(f)=A(f) \quad \text { for all } \quad f \in M^{\infty}(X) .$$
Consequently, the functional $\widehat{A}$ can be uniquely continued from $M^{\infty}(X)$ onto $L^p(X)$.

Proof: The linear functional $A$ is bounded on $\left{M^{\infty}(X),|\cdot|_{L^p(X)}\right}$ and therefore continuous. According to Theorem 6 from $\S 7$, each element $f \in L^p(X)$ possesses a sequence $\left{f_k\right}_{k=1,2, \ldots} \subset M^{\infty}(X)$ satisfying
$$\left|f_k-f\right|_{L^p(X)} \rightarrow 0 \quad \text { for } \quad k \rightarrow \infty$$
Now we define
$$\widehat{A}(f):=\lim {k \rightarrow \infty} A\left(f_k\right)$$ We immediately verify that $\widehat{A}$ has been defined independently of the sequence $\left{f_k\right}{k=1,2, \ldots}$ chosen, and that the mapping $\widehat{A}: L^p(X) \rightarrow \mathbb{R}$ is linear. Furthermore, we have
$$|\widehat{A}|=\sup {f \in L^P,|f|_p \leq 1}|\widehat{A}(f)|=\sup {f \in M^{\infty},|f|_p \leq 1}|A(f)| \leq \alpha .$$
When we consider with $\widehat{A}$ and $\widehat{B}$ two extensions of $A$ onto $L^p(X)$, we infer $\widehat{A}=$ $\widehat{B}$ on $M^{\infty}(X)$. Since the functionals $\widehat{A}$ and $\widehat{B}$ are continuous, and $M^{\infty}(X)$ lies densely in $L^p(X)$, we obtain the identity $\widehat{A}=\widehat{B}$ on $L^p(X)$.
q.e.d.

## 数学代写|偏微分方程代写partial difference equations代考|The winding number

Let us begin with the following
Definition 1. The number $k \in \mathbb{N}_0:=\mathbb{N} \cup{0}$ being prescribed, we define the set of $k$-times continuously differentiable (in the case $k \geq 1$ ) or continuous (in the case $k=0$ ) periodic complex-valued functions by the symbol
$$\Gamma_k:=\left{\varphi=\varphi(t): \mathbb{R} \rightarrow \mathbb{C} \in C^k(\mathbb{R}, \mathbb{C}): \varphi(t+2 \pi)=\varphi(t) \text { for all } t \in \mathbb{R}\right}$$
Now we note the following
Definition 2. Let the function $\varphi \in \Gamma_1$ with $\varphi(t) \neq 0$ for all $t \in \mathbb{R}$ be given. Then we define the winding number of the closed curve $\varphi(t), 0 \leq t \leq 2 \pi$ with respect to the point $z=0$ as follows:

$$W(\varphi)=W(\varphi, 0):=\frac{1}{2 \pi i} \int_0^{2 \pi} \frac{\varphi^{\prime}(t)}{\varphi(t)} d t$$
Remark: For the function $\varphi \in \Gamma_1$ we have the identity
\begin{aligned} \frac{1}{2 \pi i} \int_0^{2 \pi} \frac{\varphi^{\prime}(t)}{\varphi(t)} d t & =\frac{1}{2 \pi i} \int_0^{2 \pi} \frac{d}{d t}(\log \varphi(t)) d t \ & \left.=\frac{1}{2 \pi i} \int_0^{2 \pi} \frac{d}{d t}(\log \mid \varphi(t)) \mid+i \arg \varphi(t)\right) d t . \end{aligned}
Therefore, we obtain
$$W(\varphi)=\frac{1}{2 \pi} \int_0^{2 \pi} \frac{d}{d t}(\arg \varphi(t)) d t=\frac{1}{2 \pi}(\arg \varphi(2 \pi)-\arg \varphi(0)),$$
where we have to extend the function $\arg \varphi(t)$ along the curve continuously. The integer $W(\varphi)$ consequently describes the number of rotations (or windings) of the curve $\varphi$ about the origin.

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|§8 Bounded linear functionals on $L^p(X)$ and weak convergence

$$|A(f)| \leq \alpha|f|_{L^p(X)} \quad \text { for all } \quad f \in M^{\infty}(X)$$

$$|\widehat{A}| \leq \alpha \quad \text { and } \quad \widehat{A}(f)=A(f) \quad \text { for all } \quad f \in M^{\infty}(X) .$$

$$\left|f_k-f\right|_{L^p(X)} \rightarrow 0 \quad \text { for } \quad k \rightarrow \infty$$

$$\widehat{A}(f):=\lim {k \rightarrow \infty} A\left(f_k\right)$$我们立即验证$\widehat{A}$是独立于所选序列$\left{f_k\right}{k=1,2, \ldots}$定义的，并且映射$\widehat{A}: L^p(X) \rightarrow \mathbb{R}$是线性的。此外，我们还有
$$|\widehat{A}|=\sup {f \in L^P,|f|_p \leq 1}|\widehat{A}(f)|=\sup {f \in M^{\infty},|f|_p \leq 1}|A(f)| \leq \alpha .$$

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