## 数学代写|偏微分方程代写partial difference equations代考|MAP4341

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## 数学代写|偏微分方程代写partial difference equations代考|Continuous and compact operators

A linear mapping $T$ is continuous if and only if the image of the unit sphere under $T$ is bounded (see Appendix A.1). If this image is even relatively compact (that is, its closure is compact), then we call $T$ compact. We have thus made the following definition:

Definition 4.37 Let $E$ and $F$ be normed spaces. A linear mapping $T: E \rightarrow F$ is called compact if the following condition is satisfied: for any bounded sequence $\left(u_n\right){n \in \mathbb{N}} \subset E$ there exists a subsequence $\left(u{n_k}\right){k \in \mathbb{N}}$ such that $\left(T u{n_k}\right){k \in \mathbb{N}}$ converges in $F{\text {s }}$

Here we wish to consider such mappings in the special situation of Hilbert spaces, where we have both convergence in norm and weak convergence. We will assume throughout that $H_1$ and $H_2$ are real or complex separable Hilbert spaces.

Theorem 4.38 Let $T: H_1 \rightarrow H_2$ be linear and continuous. Then $T$ is weakly continuous, that is, $u_n \rightarrow u$ in $H_1$ implies $T u_n \rightarrow T u$ in $H_2$.

Proof Let $w \in H_2$, then $\varphi(v):=(T v, w){H_2}$ defines a continuous linear form on $H_1$. By the theorem of Riesz-Fréchet, there is thus a unique $T^* w \in H_1$ such that $(T v, w){H_2}=\left(v, T^* w\right){H_1}$ for all $v \in H_1$. If $u_n \rightarrow u$ in $H_1$, then $$\left(T u_n, w\right){H_2}=\left(u_n, T^* w\right){H_1} \rightarrow\left(u, T^* w\right){H_1}=(T u, w)_{H_2} .$$
Since $w \in \mathrm{H}_2$ was arbitrary, it follows that $T u_n \rightarrow T u$ in $H_2$.
The converse of Theorem $4.38$ is also true: every weakly continuous operator is also continuous; see Exercise 4.2. Now we can describe compact operators as being exactly those operators which “improve” convergence in the following sense.

Theorem 4.39 Let $T: H_1 \rightarrow H_2$ be linear. Then the following statements are equivalent:
(i) $T$ is compact.
(ii) $u_n \rightarrow u$ in $H_1$ implies $T u_n \rightarrow T u$ in $\mathrm{H}_2$.
For the proof we will use the following lemma which, despite looking like splitting hairs, turns out to be extremely useful.

## 数学代写|偏微分方程代写partial difference equations代考|Comments on Chapter

David Hilbert (1862-1943) was one of the pre-eminent mathematicians of the first half of last century. At the International Congress of Mathematicians (ICM) in Paris in 1900, Hilbert presented his famous 23 mathematical problems, which continue to exert considerable influence on mathematical research to this day. He was the central figure of the Göttingen school, which up until the Nazis seized power in Germany in 1933 was one of the foremost mathematical institutions worldwide. Between 1904 and 1910, Hilbert published a series of articles on integral equations, in which bilinear forms on $\ell_2$ appeared as an essential tool.

The definition of Hilbert spaces was only given in 1929, by John von Neumann. Decisive for the success of Hilbert space theory was the invention of the Lebesgue integral, in the doctoral thesis of Henri Lebesgue completed in 1902. Building on Lebesgue’s theory, Frigyes Riesz showed in 1907 that every element of $\ell_2(\mathbb{Z})$ is the sequence of Fourier coefficients of some function in $L_2(0,2 \pi)$. In the same year, Ernst Fischer (1875-1954) proved that $L_2(0,2 \pi)$ is complete, which is equivalent to Riesz’ result. This is why the name “theorem of Riesz-Fischer” is often used for the fact that $L_2(\Omega)$

is complete (see the appendix). Riesz and Maurice Fréchet (1878-1973) determined the set of all continuous linear forms on $L_2(0,1)$ independently of each other in 1907, thus proving Theorem $4.21$.

Fréchet obtained his doctorate in Paris in 1906 under Hadamard; in his dissertation, metric spaces are introduced for the first time. The theorem of Riesz-Fréchet is often known as the Riesz representation theorem; however, there is another Riesz representation theorem for measures, which will play an important role in Section 7.2.

It was Hilbert who proved a first version of the spectral theorem, Theorem 4.50. But subsequent to his works on integral equations, it was operators and not bilinear forms which took centre stage. Bounded operators on Hilbert spaces became a central concept in mathematics, and to this day operator theory remains an important area of mathematical research. In the 1930 s. quantum theory was a key driving force in the development of functional analysis. The decisive breakthrough in the mathematieal formulation of quantum physies was made by John von Neumann. who had visited Göttingen in 1926/27 and to whom the concept of unbounded operators is due (cf. Mathematische Annalen, No. 33, 1932). An unbounded self-adjoint operator (as was defined in Section 4.8) models an observable in quantum theory. The book Mathematische Grundlagen der Quantenmechanik (Mathematical Foundations of Quantum Mechanics) by John von Neumann, which appeared in 1932 in German, describes the mathematical modeling of quantum theory as is still used today, and indeed with great success. Although operators are ideal for describing equations, it is interesting that bilinear forms reached their pinnacle as a method for solving partial differential equations around 50 years after Hilbert’s work.

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|Continuous and compact operators

$$\left(T u_n, w\right) H_2=\left(u_n, T^* w\right) H_1 \rightarrow\left(u, T^* w\right) H_1$$

(i) $T$ 很紧湊。
(二) $u_n \rightarrow u$ 在 $H_1$ 暗示 $T u_n \rightarrow T u$ 在 $\mathrm{H}_2$.

## 数学代写|偏微分方程代写partial difference equations代考|Comments on Chapter

Fréchet 于 1906 年在 Hadamard 的指导下在巴黎获得 博士学位。在他的论文中，首次引入了度量空间。
Riesz-Fréchet 定理通常被称为 Riesz 表示定理；然而， 还有另一个测度的 Riesz 表示定理，它将在 $7.2$ 节中发 挥重要作用。

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