Some Main Theorems on NLS - 5. Banach Open Mapping Theorem
A map \(T:V \to W\) is said to be a open map if $T$ maps open sets in $V$ to open sets in $W$. It is a closed map if $T$ maps closed sets in $V$ to closed sets in $W$. Theorem 2.5.1 Let $V$ and $W$ be Banach spaces. Let $T:V \to W$ be linear and continuous. Then $T$ is an open map if and only if $T$ is onto. Proof. Let $U = B(0,\,1)$ be the open unit ball in $V$. We will show that there exists $..
Analysis/Functional Analysis  |  2016. 5. 10. 00:08
Some Main Theorems on NLS - 4. The Uniform Boundedness Principle
Let $(M,\,d)$ be a metric space. A set $E$ in $M$ is dense if $\mybar{E} = M$. A set $E$ is nowhere dense if $(\mybar{E})^\circ = \emptyset$. Example. $\mathbb{Q}$ is dense in $\R$ and $\mathbb{Z}$ is nowhere dense in $\R$. A set $X$ in a metric space $(M,\,d)$ is of first category if $X = \bigcup_{n=1}^\infty E_n$ where each $E_n$ is nowhere dense. A set that is not of first category if of seco..
Analysis/Functional Analysis  |  2016. 5. 9. 23:53
Some Main Theorems on NLS - 3. The space $l_p^n(\R)$, $1 \leq p \leq \infty$
Underlying space $\R^n$, let $x = (x_1,\, x_2,\, \ldots,\, x_n) \in \R^n$. Define\[ \norm{x}_p = \begin{cases} \sum_{i=1}^{n} \abs{x_i}, & \quad p=1 \\ \left( \sum_{i=1}^{n} \abs{x_i}^p \right)^{1/p}, & \quad 1
Analysis/Functional Analysis  |  2016. 5. 9. 23:36
Some Main Theorems on NLS - 2. Reflexive Banach Space
For $x \in V$. Define $f \overset{\myhat{x}}{\mapsto} f(x)$, that is, $\myhat{x}(f) = f(x)$. Then $\myhat{x} : V^* \to \F$ is linear: \[ \begin{aligned} \myhat{x}(f+g) &= (f+g)(x) = f(x) + g(x) = \myhat{x}(f) + \myhat{x}(g) \\ \myhat{x}(\lambda f) &= (\lambda f)(x) = \lambda f(x) = \lambda \myhat{x}(f). \end{aligned} \] Now, we check that $\myhat{x}$ is continuous:\[ \abs{\myhat{x}(f)} = \abs{f(..
Analysis/Functional Analysis  |  2016. 5. 9. 23:13
Some Main Theorems on NLS - 1. Hahn-Banach Extension Theorems
Let $X$ be a nonempty set. A relation ``$\leq$'' on $X$ is called a partial order if the following conditions hold:(1) Reflexive: $x \leq x$ for all $x \in X$.(2) Antisymmetric: $x \leq y$, $y \leq x$ implies $x = y$.(3) Transitive: $x \leq y$, $y \leq z$ implies $x \leq z$. Example. Let $X = \N$. Define $m \leq n$ if $m$ divides $n$. Now, let $(X,\, \leq)$ be a partially ordered set. A set $T$ ..
Analysis/Functional Analysis  |  2016. 4. 29. 05:46
Normed Linear Space - 4. The Dual Space
4. The Dual Space For two normed linear spaces $V,\, W$ over $\F$, define two spaces \[ \begin{aligned} \mathcal{L}(V,\,W) &:= \set{L:V\to W}{\text{$L$ is linear and continuous}} \\ \mathcal{B}(V,\,W) &:= \set{L:V\to W}{\text{$L$ is linear and bounded}} \\ \end{aligned} \] We have seen that $\mathcal{L}(V,\,W) = \mathcal{B}(V,\,W)$. For $L,\, L_1,\, L_2 \in \mathcal{L}(V,\,W)$, define \[ \begin{..
Analysis/Functional Analysis  |  2016. 3. 8. 06:48
Normed Linear Space - 3. Continuous Linear Transformation
3. Continuous Linear Transformation Let $(V,\, \norm{\cdot})$ and $(W,\, \norm{\cdot})$ be two normed linear spaces over the same basefield $\F$. Let $L:V \to W$ be a linear transformation (= mapping = function = operator). Recall that $L$ is continuous on $V$ if $x_n \to x$ in $(V,\, \norm{\cdot})$ implies $L(x_n) \to L(x)$ in $(W,\, \norm{\cdot})$. Theorem 1.3.1Let $L:(V,\, \norm{\cdot}) \to (..
Analysis/Functional Analysis  |  2016. 3. 8. 06:34
Normed Linear Space - 2. Finite Dimensional Normed Linear Space
2. Finite Dimensional Normed Linear Space Let $V$ be a normed linear space with dimension $n$. Let $B = \{e_1,\,\ldots,\,e_n\}$ be a basis of $V$. Fix any $x \in V$. Note that there exist $x_i \in \F$ such that \[ x = x_1e_1 + x_2e_2 + \cdots + x_ne_n. \] Now, let $\norm{\cdot}$ be the given norm on $V$. Define $\norm{\cdot}_1$ on $V$ by \[ \norm{x}_1 := \abs{x_1} +\abs{x_2} + \cdots + \abs{x_n}..
Analysis/Functional Analysis  |  2016. 3. 8. 05:59
Normed Linear Space - 1. Definitions and Examples
1. Definitions and Examples Definition 1.1.1 Let $X$ be a vector space over $\F$ ($\R$ or $\C$). A norm on $X$ is a function $\norm{\cdot} : X \to \R$ such that (1) $\norm{x} \geq 0$ for all $x \in X$. (2) $\norm{x} = 0$ if and only if $x = 0$. (3) $\norm{\lambda x} = \abs{\lambda} \norm{x}$ for all $\lambda \in \F$. (4) $\norm{x+y} \leq \norm{x} + \norm{y}$ for all $x,\, y \in X$. Then, $(X,\, ..
Analysis/Functional Analysis  |  2016. 3. 8. 03:09

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