Questions in category: 泛函分析 (Functional Analysis)

## 1. 证明 $\|u\|_{L^{\infty}(\mathbb{R})}\leqslant C\cdot\|u\|_{L^2(\mathbb{R})}^{\frac{1}{2}}\cdot\|u_x\|_{L^2(\mathbb{R})}^{\frac{1}{2}}$

Posted by haifeng on 2020-03-12 15:15:02 last update 2020-03-12 16:20:46 | Answers (0) | 收藏

$\|u\|_{L^{\infty}(\mathbb{R})}\leqslant C\cdot\|u\|_{L^2(\mathbb{R})}^{\frac{1}{2}}\cdot\|u_x\|_{L^2(\mathbb{R})}^{\frac{1}{2}}$

Remark:

## 2. 并非所有内积空间都具有正交基.

Posted by haifeng on 2016-04-05 00:02:17 last update 2016-04-05 00:02:17 | Answers (0) | 收藏

Whether all inner product spaces have an orthonormal basis? The answer is negative.

References:

https://en.wikipedia.org/wiki/Inner_product_space#Orthonormal_sequences

## 3. 若不等式 $\|\{a_n\}\|_{L^q}\leqslant A\|f\|_{L^q}$ 对所有 $f\in L^p$ 成立, 则有 $\frac{1}{p}+\frac{1}{q}\leqslant 1$.

Posted by haifeng on 2015-08-28 09:21:18 last update 2015-08-28 09:21:18 | Answers (1) | 收藏

$\|\{a_n\}\|_{L^q}\leqslant A\|f\|_{L^q}$

## 4. Sobolev 空间 $W^{k,p}$ 的定义

Posted by haifeng on 2015-06-04 11:30:35 last update 2015-12-13 21:40:44 | Answers (0) | 收藏

$W^{k,p}(\mathbb{R}):=\{f\in L^p(\mathbb{R})\mid f, f^{(1)},\ldots, f^{(k)} \in L^p(\mathbb{R})\}$

(这里考虑弱导数, 是因为了此空间是完备的, 从而是一个 Banach 空间.)

$W^{m,p}(\Omega):=\{u\in L^p(\Omega)\mid \tilde{\partial}^{\alpha}u\in L^p(\Omega),\ |\alpha|\leqslant m\}$

$\|u\|_{m,p}=\Bigl(\sum_{|\alpha|\leqslant m}\|\tilde{\partial}^{\alpha}u\|^p_{L^p(\Omega)}\Bigr)^{1/p}$

## 5. 验证 Sobolev(Coбoлeв) 空间 $\mathcal{H}^{m,p}(\Omega)$ 所定义的 $\|\cdot\|_{m,p}$ 是一个范数.

Posted by haifeng on 2012-07-07 16:39:45 last update 2012-07-07 16:46:00 | Answers (0) | 收藏

$\|u\|_{m,p}:=\biggl(\sum_{|\alpha|\leqslant m}\int_\Omega\bigl|\partial^\alpha u(x)\bigr|^p\biggr)^{\frac{1}{p}}.$

$\|u\|:=\max_{|\alpha|\leqslant k}\,\,\max_{x\in\overline{\Omega}}\bigl|\partial^\alpha u(x)\bigr|,$

References:

## 6. 证明: $\lim_{T\rightarrow+\infty}\frac{1}{\pi}\int_{-T}^{T}\frac{\sin jx}{x}dx=1$.

Posted by haifeng on 2012-07-07 16:13:51 last update 2012-07-07 16:13:51 | Answers (0) | 收藏

$\lim_{T\rightarrow+\infty}\frac{1}{\pi}\int_{-T}^{T}\frac{\sin jx}{x}dx=1.$

References:

## 7. [Thm](Banach, Steinhaus)奇点稠密原理

Posted by haifeng on 2011-09-06 09:56:43 last update 2011-09-06 15:34:39 | Answers (2) | 收藏

##### S. Banach [1] 和 H. Steinhaus 证明了奇点稠密原理(principle of condensation of singularities).

$B=\big\{x\in X\mid \varlimsup\limits_{q\rightarrow\infty}\|T_{p,q}x\|=\infty\quad\text{for all }p=1,2\ldots\big\}$

$B=\big\{x\in X\mid \varlimsup\limits_{n\rightarrow\infty}\|T_{n}x\|<\infty\big\}$

References:

Kôsaku Yosida(吉田 耕作), Functional Analysis, Sixth Edition. Springer-Verlag

[1] S. Banach, Théorie des Opérations Linéaires, Warszawa 1932.

## 8. [Thm](Weierstrass 定理)无处可微函数的存在性

Posted by haifeng on 2011-09-06 09:21:52 last update 2014-07-29 11:18:56 | Answers (0) | 收藏