diff --git a/content/Math401/Extending_thesis/Math401_R1.md b/content/Math401/Extending_thesis/Math401_R1.md index e50f83c..1eb3166 100644 --- a/content/Math401/Extending_thesis/Math401_R1.md +++ b/content/Math401/Extending_thesis/Math401_R1.md @@ -6,9 +6,239 @@ This part will cover the necessary notations and definitions for the remaining parts of the recollection. -### Notations of Hilbert space +### Notations of Linear algebra -A Hilbert space is a vector space equipped with an inner product. +#### Definition of vector space + +[link to vector space](../../Math429/Math429_L1#definition-1.20) + +A vector space over $\mathbb{f}$ is a set $V$ along with two operators $v+w\in V$ for $v,w\in V$, and $\lambda \cdot v$ for $\lambda\in \mathbb{F}$ and $v\in V$ satisfying the following properties: + +* Commutativity: $\forall v, w\in V,v+w=w+v$ +* Associativity: $\forall u,v,w\in V,(u+v)+w=u+(v+w)$ +* Existence of additive identity: $\exists 0\in V$ such that $\forall v\in V, 0+v=v$ +* Existence of additive inverse: $\forall v\in V, \exists w \in V$ such that $v+w=0$ +* Existence of multiplicative identity: $\exists 1 \in \mathbb{F}$ such that $\forall v\in V,1\cdot v=v$ +* Distributive properties: $\forall v, w\in V$ and $\forall a,b\in \mathbb{F}$, $a\cdot(v+w)=a\cdot v+ a\cdot w$ and $(a+b)\cdot v=a\cdot v+b\cdot v$ + +#### Definition of inner product + +[link to inner product](../../Math429/Math429_L25#definition-6.2) + +An inner product is a bilinear function $\langle,\rangle:V\times V\to \mathbb{F}$ satisfying the following properties: + +* Positivity: $\langle v,v\rangle\geq 0$ +* Definiteness: $\langle v,v\rangle=0\iff v=0$ +* Additivity: $\langle u+v,w\rangle=\langle u,w\rangle+\langle v,w\rangle$ +* Homogeneity: $\langle \lambda u, v\rangle=\lambda\langle u,v\rangle$ +* Conjugate symmetry: $\langle u,v\rangle=\overline{\langle v,u\rangle}$ + +
+Examples of inner product + +Let $V=\mathbb{R}^n$. + +The dot product is defined by + +$$ +\langle u,v\rangle=u_1v_1+u_2v_2+\cdots+u_nv_n +$$ + +is an inner product. + +--- + +Let $V=L^2(\mathbb{R}, \lambda)$, where $\lambda$ is the Lebesgue measure. $f,g:\mathbb{R}\to \mathbb{C}$ are complex-valued square integrable functions. + +The Hermitian inner product is defined by +$$ +\langle f,g\rangle=\int_\mathbb{R} \overline{f(x)}g(x) d\lambda(x) +$$ + +is an inner product. + +--- + +Let $A,B$ be two linear transformation on $\mathbb{R}^n$. + +The Hilbert-Schmidt inner product is defined by + +$$ +\langle A,B\rangle=\operatorname{Tr}(A^*B)=\sum_{i=1}^n \sum_{j=1}^n \overline{a_{ij}}b_{ij} +$$ + +is an inner product. + +
+ +#### Definition of inner product space + +A inner product space is a vector space equipped with an inner product. + +#### Definition of completeness + +[link to completeness](../../Math4111/Math4111_L17#definition-312) + +Note that every inner product space is a metric space. + +Let $X$ be a metric space. We say $X$ is **complete** if every Cauchy sequence (that is, a sequence such that $\forall \epsilon>0, \exists N$ such that $\forall m,n\geq N, d(p_m,p_n)<\epsilon$) in $X$ converges. + +#### Definition of Hilbert space + +A Hilbert space is a complete inner product space. + +#### Motivation of Tensor product + +Recall from the traditional notation of product space of two vector spaces $V$ and $W$, that is, $V\times W$, is the set of all ordered pairs $(v,w)$ where $v\in V$ and $w\in W$. + +The space has dimension $\dim V+\dim W$. + +We want to define a vector space with notation of multiplication of two vectors from different vector spaces. + +That is + +$$ +(v_1+v_1)\otimes w=(v_1\otimes w)+(v_2\otimes w)\text{ and } v\otimes (w_1+w_2)=(v\otimes w_1)+(v\otimes w_2) +$$ + +and enables scalar multiplication by + +$$ +\lambda (v\otimes w)=(\lambda v)\otimes w=v\otimes (\lambda w) +$$ + +And we wish to build a way associates the basis of $V$ and $W$ to the basis of $V\otimes W$. That makes the tensor product a vector space with dimension $\dim V\times \dim W$. + +#### Definition of linear functional + +> [!TIP] +> +> Note the difference between a linear functional and a linear map. +> +> A generalized linear map is a function $f:V\to W$ satisfying the condition +> +> 1. $f(u+v)=f(u)+f(v)$ +> 2. $f(\lambda v)=\lambda f(v)$ + +A linear functional is a linear map from $V$ to $\mathbb{F}$. + +#### Definition of bilinear functional + +A bilinear functional is a bilinear function $\beta:V\times W\to \mathbb{F}$ satisfying the condition that $v\to \beta(v,w)$ is a linear functional for all $w\in W$ and $w\to \beta(v,w)$ is a linear functional for all $v\in V$. + +The vector space of all bilinear functionals is denoted by $\mathcal{B}(V,W)$. + +#### Definition of tensor product + +Let $V,W$ be two vector spaces. + +Let $V'$ and $W'$ be the dual spaces of $V$ and $W$, respectively, that is $V'=\{\psi:V\to \mathbb{F}\}$ and $W'=\{\phi:W\to \mathbb{F}\}$, $\psi, \phi$ are linear functionals. + +The tensor product of vectors $v\in V$ and $w\in W$ is the bilinear functional defined by $\forall (\psi,\phi)\in V'\times W'$ given by the notation + +$$ +(v\otimes w)(\psi,\phi)\coloneqq\psi(v)\phi(w) +$$ + +The tensor product of two vector spaces $V$ and $W$ is the vector space $\mathcal{B}(V',W')$ + +Notice that the basis of such vector space is the linear combination of the basis of $V'$ and $W'$, that is, if $\{e_i\}$ is the basis of $V'$ and $\{f_j\}$ is the basis of $W'$, then $\{e_i\otimes f_j\}$ is the basis of $\mathcal{B}(V',W')$. + +That is, every element of $\mathcal{B}(V',W')$ can be written as a linear combination of the basis. + +Since $\{e_i\}$ and $\{f_j\}$ are bases of $V'$ and $W'$, respectively, then we can always find a set of linear functionals $\{\phi_i\}$ and $\{\psi_j\}$ such that $\phi_i(e_j)=\delta_{ij}$ and $\psi_j(f_i)=\delta_{ij}$. + +Here $\delta_{ij}=\begin{cases} +1 & \text{if } i=j \\ +0 & \text{otherwise} +\end{cases}$ is the Kronecker delta. + +$$ +V\otimes W=\left\{\sum_{i=1}^n \sum_{j=1}^m a_{ij} \phi_i(v)\psi_j(w): \phi_i\in V', \psi_j\in W'\right\} +$$ + +Note that $\sum_{i=1}^n \sum_{j=1}^m a_{ij} \phi_i(v)\psi_j(w)$ is a bilinear functional that maps $V'\times W'$ to $\mathbb{F}$. + +This enables basis free construction of vector spaces with proper multiplication and scalar multiplication. + +This vector space is equipped with the unique inner product $\langle v\otimes w, u\otimes x\rangle_{V\otimes W}$ defined by + +$$ +\langle v\otimes w, u\otimes x\rangle=\langle v,u\rangle_V\langle w,x\rangle_W +$$ + +In practice, we ignore the subscript of the vector space and just write $\langle v\otimes w, u\otimes x\rangle=\langle v,u\rangle\langle w,x\rangle$. + +> [!NOTE] +> +> All those definitions and proofs can be found in Linear Algebra Done Right by Sheldon Axler. + +### Notations in measure theory + +#### Definition of Sigma algebra + +[link to measure theory](../../Math4121/Math4121_L25#definition-of-sigma-algebra) + +A collection of sets $\mathcal{A}$ is called a sigma-algebra if it satisfies the following properties: + +1. $\emptyset \in \mathcal{A}$ +2. If $\{A_j\}_{j=1}^\infty \subset \mathcal{A}$, then $\bigcup_{j=1}^\infty A_j \in \mathcal{A}$ +3. If $A \in \mathcal{A}$, then $A^c \in \mathcal{A}$ + +#### Definition of Measure + +A measure is a function $v:\mathcal{A}\to \mathbb{R}$ satisfying the following properties: + +1. $v(\emptyset)=0$ +2. If $\{A_j\}_{j=1}^\infty \subset \mathcal{A}$ are pairwise disjoint, then $v(\bigcup_{j=1}^\infty A_j)=\sum_{j=1}^\infty v(A_j)$ (countable additivity) +3. If $A\in \mathcal{A}$, then $v(A)\geq 0$ (non-negativity) + +
+Examples of measure + +The [Borel measure on $\mathbb{R}$](../../Math4121/Math4121_L25#definition-of-borel-measure) is the collection of all closed, open, and half-open intervals with $m(U)=\ell(U)$ for any open set $U$. + +The [Lebesgue measure on $\mathbb{R}$](../../Math4121/Math4121_L27#definition-of-lebesgue-measure) is the collection of all Lebesgue measurable sets with $m_i=\sup_{K\text{ closed},K\subseteq S}m(K)$ and $m_e=\inf_{U\text{ open},S\subseteq U}m(U)$. and $m(S)=m_e(S)=m_i(S)$ for any Lebesgue measurable set $S$. + +
+ +#### Definition of Probability measure + +Let $\mathscr{F}$ be a sigma-algebra on a set $\Omega$. A probability measure is a function $P:\mathscr{F}\to [0,1]$ satisfying the following properties: + +1. $P(\Omega)=1$ +2. $P$ is a measure on $\mathscr{F}$ + +#### Definition of Measurable space + +A measurable space is a pair $(X, \mathscr{B}, v)$, where $X$ is a set and $\mathscr{B}$ is a sigma-algebra on $X$. + +In some literatures, $\mathscr{B}$ is ignored and we only denote it as $(X, v)$. + +
+Examples of measurable space + +Let $\Omega$ be arbitrary set. + +Let $\mathscr{B}(\mathbb{C})$ be the Borel sigma-algebra on $\mathbb{C}$ generated from rectangles over complex plane with real number axes and $\lambda$ be the Lebesgue measure associated with it. + +Let $\mathscr{F}$ be the set of square integrable, that is, + +$$ +\int_\Omega |f(x)|^2 d\lambda(x)<\infty +$$ + +complex-valued functions on $\Omega$, that is, $f:\Omega\to \mathbb{C}$. + +Then the measurable space $(\Omega, \mathscr{B}(\mathbb{C}), \lambda)$ is a measurable space. We usually denote this as $L^2(\Omega, \mathscr{B}(\mathbb{C}), \lambda)$. + +If $\Omega=\mathbb{R}$, then we denote such measurable space as $L^2(\mathbb{R}, \lambda)$. + +
+ +#### Probability space + +A probability space is a triple $(\Omega, \mathscr{F}, P)$, where $\Omega$ is a set, $\mathscr{F}$ is a sigma-algebra on $\Omega$, and $P$ is a probability measure on $\mathscr{F}$. ### Lipschitz function @@ -28,7 +258,7 @@ That basically means that the function $f$ should not change the distance betwee Basic definitions -### $SO(n)$ +#### $SO(n)$ The special orthogonal group $SO(n)$ is the set of all **distance preserving** linear transformations on $\mathbb{R}^n$. diff --git a/content/Math401/Extending_thesis/Math401_R2.md b/content/Math401/Extending_thesis/Math401_R2.md index efe7301..12b0cd7 100644 --- a/content/Math401/Extending_thesis/Math401_R2.md +++ b/content/Math401/Extending_thesis/Math401_R2.md @@ -270,9 +270,26 @@ Not very edible for undergraduates. #### Definition of m-manifold -An $m$-manifold is a [Hausdorff space](../../Math4201/Math4201_L9#hausdorff-space) $X$ with a countable basis such that each point of $x$ of $X$ has a neighborhood [homeomorphic](../../Math4201/Math4201_L10#definition-of-homeomorphism) to an open subset of $\mathbb{R}^m$. +An $m$-manifold is a [Hausdorff space](../../Math4201/Math4201_L9#hausdorff-space) $X$ with a **countable basis** (second countable) such that each point of $x$ of $X$ has a neighborhood [homeomorphic](../../Math4201/Math4201_L10#definition-of-homeomorphism) to an open subset of $\mathbb{R}^m$. -Example is trivial that 1-manifold is a curve and 2-manifold is a surface. +
+Example of second countable space + +Let $X=\mathbb{R}$ and $\mathcal{B}=\{(a,b)|a,b\in \mathbb{R},a + +
+Example of manifold + +1-manifold is a curve and 2-manifold is a surface. + +
#### Theorem of imbedded space @@ -280,10 +297,51 @@ If $X$ is a compact $m$-manifold, then $X$ can be imbedded in $\mathbb{R}^n$ for This theorem might save you from imagining abstract structures back to real dimension. Good news, at least you stay in some real numbers. -### Smooth manifold +### Smooth manifolds and Lie groups > This section is waiting for the completion of book Introduction to Smooth Manifolds by John M. Lee. +#### Partial derivatives + +Let $U\subseteq \mathbb{R}^n$ and $f:U\to \mathbb{R}^n$ be a map. + +For any $a=(a_1,\cdots,a_n)\in U$, $j\in \{1,\cdots,n\}$, the $j$-th partial derivative of $F$ at $a$ is defined as + +$$ +\begin{aligned} +\frac{\partial f}{\partial x_j}(a)&=\lim_{h\to 0}\frac{f(a_1,\cdots,a_j+h,\cdots,a_n)-f(a_1,\cdots,a_j,\cdots,a_n)}{h} \\ +&=\lim_{h\to 0}\frac{f(a+he_j)-f(a)}{h} +\end{aligned} +$$ + +#### Continuously differentiable maps + +Let $U\subseteq \mathbb{R}^n$ and $f:U\to \mathbb{R}^n$ be a map. + +If for any $j\in \{1,\cdots,n\}$, the $j$-th partial derivative of $f$ is continuous at $a$, then $f$ is continuously differentiable at $a$. + +If $\forall a\in U$, $\frac{\partial f}{\partial x_j}$ exists and is continuous at $a$, then $f$ is continuously differentiable on $U$. or $C^1$ map. (Note that $C^0$ map is just a continuous map.) + +#### Smooth maps + +A function $f:U\to \mathbb{R}^n$ is smooth if it is of class $C^k$ for every $k\geq 0$ on $U$. Such function is called a diffeomorphism if it is also a **bijection** and its **inverse is also smooth**. + +#### Charts + +Let $M$ be a smooth manifold. A **chart** is a pair $(U,\phi)$ where $U\subseteq M$ is an open subset and $\phi:U\to \hat{U}\subseteq \mathbb{R}^n$ is a homeomorphism (a continuous bijection map and its inverse is also continuous). + +If $p\in U$ and $\phi(p)=0$, then we say that $p$ is the origin of the chart $(U,\phi)$. + +#### Atlas + +Let $M$ be a smooth manifold. An **atlas** is a collection of charts $\mathcal{A}=\{(U_\alpha,\phi_\alpha)\}_{\alpha\in I}$ such that $M=\bigcup_{\alpha\in I} U_\alpha$. + +An atlas is said to be **smooth** if the transition maps $\phi_\alpha\circ \phi_\beta^{-1}:\phi_\beta(U_\alpha\cap U_\beta)\to \phi_\alpha(U_\alpha\cap U_\beta)$ are smooth for all $\alpha, \beta\in I$. + +#### Smooth manifold + +A smooth manifold is a pair $(M,\mathcal{A})$ where $M$ is a topological manifold and $\mathcal{A}$ is a smooth atlas. + ### Riemannian manifolds A Riemannian manifold is a smooth manifold equipped with a **Riemannian metric**, which is a smooth assignment of an inner product to each tangent space $T_pM$ of the manifold. diff --git a/content/Math4121/Math4121_L17.md b/content/Math4121/Math4121_L17.md index 836af8a..bd213f1 100644 --- a/content/Math4121/Math4121_L17.md +++ b/content/Math4121/Math4121_L17.md @@ -24,7 +24,8 @@ Given a set $S$, the power set of $S$, denoted $\mathscr{P}(S)$ or $2^S$, is the Cardinality of $2^S$ is not equal to the cardinality of $S$. -Proof: +
+Proof of Cantor's Theorem Assume they have the same cardinality, then $\exists \psi: S \to 2^X$ which is one-to-one and onto. (this function returns a subset of $S$) @@ -38,7 +39,7 @@ If $b\in T$, then by definition of $T$, $b \notin \psi(b)$, but $\psi(b) = T$, w If $b \notin T$, then $b \in \psi(b)$, which is also a contradiction since $b\in T$. Therefore, $2^S$ cannot have the same cardinality as $S$. -QED +
### Back to Hankel's Conjecture diff --git a/content/Math4121/Math4121_L18.md b/content/Math4121/Math4121_L18.md index 7c18629..f9e6804 100644 --- a/content/Math4121/Math4121_L18.md +++ b/content/Math4121/Math4121_L18.md @@ -12,13 +12,16 @@ By modifying this example, we can find similar with any outer content between 0 $S\subseteq[0,1]$ is perfect if $S=S'$. -Example: +
+Examples of perfect set - $[0,1]$ is perfect - perfect sets are closed - Finite collection of points is not perfect because they do not have limit points. - perfect sets are uncountable (no countable sets can be perfect) +
+ #### Middle third Cantor set We construct the set by removing the middle third of the interval. @@ -49,7 +52,8 @@ $$ $C$ is perfect and nowhere dense, and outer content is 0. -Proof: +
+Proof (i) $c_e(C)=0$ @@ -70,3 +74,4 @@ It is sufficient to show $C$ contains no intervals. Any open intervals has a real number with 1 in it's base 3 decimal expansion (proof in homework) _take some interval in $(a,b)$ we can change the digits that is small enough and keep the element still in the set_ +
\ No newline at end of file diff --git a/content/Math4121/Math4121_L19.md b/content/Math4121/Math4121_L19.md index 07a1ca6..c1f767b 100644 --- a/content/Math4121/Math4121_L19.md +++ b/content/Math4121/Math4121_L19.md @@ -44,11 +44,12 @@ The outer content of $SVC(n)$ is $\frac{n-3}{n-2}$. If $S\subseteq T$, then $c_e(S)\leq c_e(T)$. -Proof: +
+Proof of Monotonicity of outer content If $C$ is cover of $T$, then $S\subseteq T\subseteq C$, so $C$ is a cover of $S$. Since $c_e(s)$ takes the inf over a larger set that $c_e(T)$, $c_e(S) \leq c_e(T)$. -QED +
#### Theorem Osgood's Lemma diff --git a/content/Math4121/Math4121_L20.md b/content/Math4121/Math4121_L20.md index b78b287..d09dece 100644 --- a/content/Math4121/Math4121_L20.md +++ b/content/Math4121/Math4121_L20.md @@ -30,7 +30,8 @@ If $S=\bigcup_{n=1}^{\infty} I_n$, $T=\bigcup_{n=1}^{\infty} J_n$, where $I_n$ a Let $S$ be a closed, bounded set in $\mathbb{R}$, and $S_1\subseteq S_2\subseteq \ldots$, and $S=\bigcup_{n=1}^{\infty} S_n$. Then $\lim_{k\to\infty} c_e(S_k)=c_e(S)$. -Proof: +
+Proof of Osgood's Lemma Trivial that $c_e(S_k)\leq c_e(S)$. @@ -70,7 +71,7 @@ c_e(S)&\leq c_e(U)\\ \end{aligned} $$ -QED +
### Convergence Theorems for sequences of functions @@ -96,7 +97,8 @@ $$ \lim_{n\to\infty}\int_a^b f_n(x)\ dx=\int_a^b f(x)\ dx $$ -Proof: +
+Proof of Arzela-Osgood Theorem (incomplete) Define $\Gamma_{\alpha}=\{x:\forall m\in \mathbb{N} \textup{ and }\forall \delta>0, \exists n\geq m \textup{ s.t. } |y-x|<\delta \textup{ and } |f_n(y)-f_m(y)|>\alpha\}$. @@ -105,3 +107,4 @@ _$\Gamma_{\alpha}$ is the negation of $(\alpha,\delta)$ definition of limit._ $\Gamma_{\alpha}$ is closed and nowhere dense. Continue on next lecture. +
\ No newline at end of file diff --git a/content/Math4121/Math4121_L21.md b/content/Math4121/Math4121_L21.md index c9e2cdc..fa23e34 100644 --- a/content/Math4121/Math4121_L21.md +++ b/content/Math4121/Math4121_L21.md @@ -20,7 +20,8 @@ $$ Fact: $\Gamma_{\alpha}$ is closed and nowhere dense. -Proof: +
+Proof Without loss of generality, we can assume $f=0$. Given any $\alpha > 0$, $\exists N$ such that @@ -70,5 +71,5 @@ This implies $\ell(P_2)\leq \frac{\alpha}{4B}$. Continue on Friday. -QED +
diff --git a/content/Math4121/Math4121_L22.md b/content/Math4121/Math4121_L22.md index 42f310b..438a10e 100644 --- a/content/Math4121/Math4121_L22.md +++ b/content/Math4121/Math4121_L22.md @@ -2,7 +2,8 @@ ## Continue on Arzela-Osgood Theorem -Proof: +
+Proof continuation of Arzela-Osgood Theorem Part 2: Control the integral on $\mathcal{U}$ @@ -40,7 +41,7 @@ $$ $\forall N\geq K$. -QED +
### Baire Category Theorem @@ -50,7 +51,8 @@ Nowhere dense sets can be large, but they canot cover an open (or closed) interv An open interval cannot be covered by a countable union of nowhere dense sets. -Proof: +
+Proof Suppose $(0,1)\subset \bigcup_{n=1}^\infty S_n$ where each $S_n$ is nowhere dense. In particular, $\exists I_1$ closed interval such that $I_1\subset (0,1)$ and $I_1\cap S_1=\emptyset$. @@ -62,7 +64,7 @@ Then $x\in (0,1)$ and $x\notin \bigcup_{n=1}^\infty S_n$. Contradiction with the assumption that $(0,1)\subset \bigcup_{n=1}^\infty S_n$. -QED +
#### Definition First Category @@ -72,13 +74,14 @@ A countable union of nowhere dense sets is called a set of **first category**. Complement of a set of first category in $\mathbb{R}$ is dense in $\mathbb{R}$. -Proof: +
+Proof We need to show that for every interval $I$, $\exists x\in I\cap S^c$. ($\exists x\in I$ and $x\notin S$) This is equivalent to the Baire Category Theorem. -QED +
Recall a function is pointwise discontinuous if $\mathcal{C}=\{c\in [a,b]: f\text{ is continuous at } c\}$ is dense in $[a,b]$. @@ -88,7 +91,8 @@ $\mathcal{D}=[a,b]\setminus \mathcal{C}$ is called the set of points of disconti $f$ is pointwise discontinuous if and only if $\mathcal{D}$ is of first category. -Proof: +
+Proof Part 1: If $\mathcal{D}$ is of first category, then $f$ is pointwise discontinuous. @@ -104,13 +108,14 @@ Let $I\subseteq [a,b]$ so $\exists c\in \mathcal{C}\cap I$. So by definition of Thus, $P_k$ is nowhere dense. -QED +
#### Corollary 4.10 Let $\{f_n\}$ be a sequence of pointwise discontinuous functions. The set of points at which all $f_n$ are simultaneously continuous is dense (it's also uncountable). -Proof: +
+Proof $$ \bigcap_{n=1}^\infty \mathcal{C}_n=\left(\bigcup_{n=1}^\infty \mathcal{D}_n\right)^c @@ -118,4 +123,4 @@ $$ The complement of a set of first category is dense. -QED +
diff --git a/content/Math4121/Math4121_L23.md b/content/Math4121/Math4121_L23.md index 5789b11..d419fa6 100644 --- a/content/Math4121/Math4121_L23.md +++ b/content/Math4121/Math4121_L23.md @@ -110,7 +110,8 @@ $$ So $S$ is Jordan measurable if and only if $c_e(\partial S)=0$. -Proof: +
+Proof Let $\epsilon > 0$, and $\{R_j\}_{j=1}^N$ be an open cover of $\partial S$. such that $\sum_{j=1}^N \text{vol}(R_j) < c_e(\partial S)+\frac{\epsilon}{2}$. @@ -136,4 +137,4 @@ If $\eta$ is small enough (depends on $\delta$), then $\mathcal{C}_\eta=\{Q\in K Suppose $\exists x\in S$ but not in $\mathcal{C}_\eta$. Then $x$ is closed to $\partial S$ so in some $Q_j$. (This proof is not rigorous, but you get the idea. Also not clear in book actually.) -EOP +
diff --git a/content/Math4121/Math4121_L24.md b/content/Math4121/Math4121_L24.md index b1bd246..3052aa5 100644 --- a/content/Math4121/Math4121_L24.md +++ b/content/Math4121/Math4121_L24.md @@ -14,7 +14,8 @@ $$ where $\partial S$ is the boundary of $S$ and $c_e(\partial S)=0$. -Example: +
+Examples for Jordan measurable 1. $S=\mathbb{Q}\cap [0,1]$ is not Jordan measurable. @@ -56,6 +57,8 @@ So $c_e(SVC(4))=\frac{1}{2}$. > General formula for $c_e(SVC(n))=\frac{n-3}{n-2}$, and since $SVC(n)$ is nowhere dense, $c_i(SVC(n))=0$. +
+ ### Additivity of Content Recall that outer content is sub-additive. Let $S,T\subseteq \mathbb{R}^n$ be disjoint. @@ -80,7 +83,8 @@ $$ c(\bigcup_{i=1}^N S_i)=\sum_{i=1}^N c(S_i) $$ -Proof: +
+Proof $$ \begin{aligned} @@ -96,7 +100,7 @@ $$ c(\bigcup_{i=1}^N S_i)=\sum_{i=1}^N c(S_i) $$ -QED +
##### Failure for countable additivity for Jordan content diff --git a/content/Math4121/Math4121_L25.md b/content/Math4121/Math4121_L25.md index 28a36a0..e99aa0c 100644 --- a/content/Math4121/Math4121_L25.md +++ b/content/Math4121/Math4121_L25.md @@ -48,7 +48,8 @@ The Borel sets are Borel measurable. (proof in the following lectures) -Examples: +
+Examples for Borel measurable 1. Let $S=\{x\in [0,1]: x\in \mathbb{Q}\}$ @@ -62,6 +63,8 @@ Since $c_e(SVC(4))=\frac{1}{2}$ and $c_i(SVC(4))=0$, it is not Jordan measurable $S$ is Borel measurable with $m(S)=\frac{1}{2}$. (use setminus and union to show) +
+ #### Proposition 5.3 Let $\mathcal{B}$ be the Borel sets in $\mathbb{R}$. Then the cardinality of $\mathcal{B}$ is $2^{\aleph_0}=\mathfrak{c}$. But the cardinality of the set of Jordan measurable sets is $2^{\mathfrak{c}}$. diff --git a/content/Math4121/Math4121_L27.md b/content/Math4121/Math4121_L27.md index 1ff3c3e..6cd7d59 100644 --- a/content/Math4121/Math4121_L27.md +++ b/content/Math4121/Math4121_L27.md @@ -14,7 +14,7 @@ where $I_j$ is an open interval 1. $m_e(I)=\ell(I)$ 2. Countably sub-additive: $m_e\left(\bigcup_{n=1}^\infty S_n\right)\leq \sum_{n=1}^\infty m_e(S_n)$ (Prove today) -3. does not repect complementation (Build in to Borel measure) +3. does not respect complementation (Build in to Borel measure) Why does Jordan content respect complementation? @@ -64,7 +64,8 @@ $$ m_e\left(\bigcup_{n=1}^\infty S_n\right)\leq\sum_{n=1}^\infty m(S_n) $$ -Proof: +
+Proof Let $\epsilon>0$ and for each $j$, let $\{I_{i,j}\}_{i=1}^\infty$ be a cover of $S_j$ s.t. @@ -84,21 +85,22 @@ $$ m_e\left(\bigcup_{j=1}^\infty S_j\right)\leq\sum_{j=1}^\infty m_e(S_j)=\sum_{j=1}^\infty m(S_j) $$ -QED +
-#### Corollary +#### Corollary: inner measure is always less than or equal to outer measure $$ m_i(S)\leq m_e(S) $$ -Proof: +
+Proof $$ m_i(S)=m(I)-m_e(I\setminus S)\leq m(I)-m_i(I\setminus S)=m_e(S) $$ -QED +
### Caratheodory's Criterion @@ -110,7 +112,8 @@ $$ m_e\left(S\cap \left(\bigcup_{j=1}^\infty I_j\right)\right)=m_e\left(\bigcup_{j=1}^\infty (S\cap I_j)\right)=\sum_{j=1}^\infty m_e(S\cap I_j) $$ -Proof: +
+Proof For each $j$, let $\{J_i\}_{i=1}^\infty$ be a cover of $S\cap \left(\bigcup_{j=1}^\infty I_j\right)$ such that $\sum_{i=1}^\infty \ell(J_i) #### Theorem 5.6 (Caratheodory's Criterion) diff --git a/content/Math4121/Math4121_L28.md b/content/Math4121/Math4121_L28.md index 50f4e0b..3b80428 100644 --- a/content/Math4121/Math4121_L28.md +++ b/content/Math4121/Math4121_L28.md @@ -31,7 +31,8 @@ $$ > $$m_e\left(S\cap \bigcup_{j=1}^{\infty} I_j\right) = \sum_{j=1}^{\infty} m_e(S\cap I_j)$$ > Proved on Friday -Proof: +
+Proof $\implies$ If Lebesgue criterion holds for $S$, then for any $X$ of finite outer measure, @@ -72,7 +73,7 @@ m_e(X)&\leq m_e(X\cap S)+m_e(S^c\cap X)\\ \end{aligned} $$ -QED +
### Revisit Borel's criterion @@ -88,7 +89,8 @@ $$ m_e(S)=\sum_{j=1}^{\infty} m_e(S_j) $$ -Proof: +
+Proof First we prove $m_e(\bigcup_{j=1}^{\infty} S_j)=\sum_{j=1}^{\infty} m(S_j)$ by induction. @@ -116,16 +118,17 @@ Therefore, $\sum_{j=1}^{\infty} m(S_j)\leq m_e(S)\leq \sum_{j=1}^{\infty} m(S_j) So $S$ is measurable. -QED +
#### Proposition 5.9 (Preview) Any finite union (and intersection) of measurable sets is measurable. -Proof: +
+Proof Let $S_1, S_2$ be measurable sets. We prove by verifying the Caratheodory's criteria for $S_1\cup S_2$. -QED +
diff --git a/content/Math4121/Math4121_L29.md b/content/Math4121/Math4121_L29.md index b9fd2f8..358eb0b 100644 --- a/content/Math4121/Math4121_L29.md +++ b/content/Math4121/Math4121_L29.md @@ -34,7 +34,8 @@ Towards proving $\mathfrak{M}$ is closed under countable unions: Any finite union/intersection of Lebesgue measurable sets is Lebesgue measurable. -Proof: +
+Proof Suppose $S_1, S_2$ is a measurable, and we need to show that $S_1\cup S_2$ is measurable. Given $X$, need to show that @@ -61,13 +62,14 @@ $$ by measurability of $S_1$ again. -QED +
#### Theorem 5.10 (Countable union/intersection of Lebesgue measurable sets is Lebesgue measurable) Any countable union/intersection of Lebesgue measurable sets is Lebesgue measurable. -Proof: +
+Proof Let $\{S_j\}_{j=1}^{\infty}\subset\mathfrak{M}$. Definte $T_j=\bigcup_{k=1}^{j}S_k$ such that $T_{j-1}\subset T_j$ for all $j$. @@ -109,7 +111,7 @@ Therefore, $m_e(X\cap S)=m_e(X)$. Therefore, $S$ is measurable. -QED +
#### Corollary from the proof diff --git a/content/Math4121/Math4121_L30.md b/content/Math4121/Math4121_L30.md index 631d363..ed10fc8 100644 --- a/content/Math4121/Math4121_L30.md +++ b/content/Math4121/Math4121_L30.md @@ -20,7 +20,8 @@ $$ m_i(S)=\sup_{K\text{ closed},K\subseteq S}m(K) $$ -Proof: +
+Proof Inner regularity: @@ -34,7 +35,7 @@ $$ So $m_i(S) We can approximate $m(S)$ from outside by open sets. If we are just concerned with "approximating" $m(S)$, we can use finite union of intervals. @@ -58,7 +59,8 @@ $$ where $U=\bigcup_{j =1}^n I_j$. -Proof: +
+Proof Let $\epsilon>0$ and $m(V) Recall $\{T_j\}_{j=1}^\infty$ are disjoint measurable sets. Then $T=\bigcup_{j=1}^\infty T_j$ is measurable and diff --git a/content/Math429/Math429_L25.md b/content/Math429/Math429_L25.md index 953f3e0..790f64d 100644 --- a/content/Math429/Math429_L25.md +++ b/content/Math429/Math429_L25.md @@ -23,17 +23,17 @@ Some properties #### Definition 6.2 -An inner product $<,>:V\times V\to \mathbb{F}$ +An inner product $\langle,\rangle:V\times V\to \mathbb{F}$ -Positivity: $\geq 0$ +Positivity: $\langle v,v\rangle\geq 0$ -Definiteness: $=0\iff v=0$ +Definiteness: $\langle v,v\rangle=0\iff v=0$ -Additivity: $=+$ +Additivity: $\langle u+v,w\rangle=\langle u,w\rangle+\langle v,w\rangle$ -Homogeneity: $<\lambda u, v>=\lambda$ +Homogeneity: $\langle \lambda u, v\rangle=\lambda\langle u,v\rangle$ -Conjugate symmetry: $=\overline{}$ +Conjugate symmetry: $\langle u,v\rangle=\overline{\langle v,u\rangle}$ Note: the dot product on $\mathbb{R}^n$ satisfies these properties @@ -43,39 +43,39 @@ $V=C^0([-1,-])$ $L_2$ - inner product. -$=\int^1_{-1} f\cdot g$ +$\langle f,g\rangle=\int^1_{-1} f\cdot g$ -$=\int ^1_{-1}f^2\geq 0$ +$\langle f,f\rangle=\int ^1_{-1}f^2\geq 0$ -$=+$ +$\langle f+g,h\rangle=\langle f,h\rangle+\langle g,h\rangle$ -$<\lambda f,g>=\lambda$ +$\langle \lambda f,g\rangle=\lambda\langle f,g\rangle$ -$=\int^1_{-1} f\cdot g=\int^1_{-1} g\cdot f=$ +$\langle f,g\rangle=\int^1_{-1} f\cdot g=\int^1_{-1} g\cdot f=\langle g,f\rangle$ The result is in real vector space so no conjugate... #### Theorem 6.6 -For $<,>$ an inner product +For $\langle,\rangle$ an inner product -(a) Fix $V$, then the map given by $u\mapsto $ is a linear map (Warning: if $\mathbb{F}=\mathbb{C}$, then $u\mapsto$ is not linear). +(a) Fix $V$, then the map given by $u\mapsto \langle u,v\rangle$ is a linear map (Warning: if $\mathbb{F}=\mathbb{C}$, then $u\mapsto\langle u,v\rangle$ is not linear). -(b,c) $<0,v>==0$ +(b,c) $\langle 0,v\rangle=\langle v,0\rangle=0$ -(d) $=+$ (second terms are additive.) +(d) $\langle u,v+w\rangle=\langle u,v\rangle+\langle u,w\rangle$ (second terms are additive.) -(e) $=\bar{\lambda}$ +(e) $\langle u,\lambda v\rangle=\bar{\lambda}\langle u,v\rangle$ #### Definition 6.4 -An **inner product space** is a pair of vector space and inner product on it. $(v,<,>)$. In practice, we will say "$V$ is an inner product space" and treat $V$ as the vector space. +An **inner product space** is a pair of vector space and inner product on it. $(v,\langle,\rangle)$. In practice, we will say "$V$ is an inner product space" and treat $V$ as the vector space. For the remainder of the chapter. $V,W$ are inner product vector spaces... #### Definition 6.7 -For $v\in V$ the **norm of $V$** is given by $||v||:=\sqrt{}$ +For $v\in V$ the **norm of $V$** is given by $||v||:=\sqrt{\langle v,v\rangle}$ #### Theorem 6.9 @@ -86,13 +86,13 @@ Suppose $v\in V$. Proof: -$||\lambda v||^2=<\lambda v,\lambda v> =\lambda=\lambda\bar{\lambda}$ +$||\lambda v||^2=\langle \lambda v,\lambda v\rangle =\lambda\langle v,\lambda v\rangle=\lambda\bar{\lambda}\langle v,v\rangle$ -So $|\lambda|^2 =|\lambda|^2||v||^2$, $||\lambda v||=|\lambda|\ ||v||$ +So $|\lambda|^2 \langle v,v\rangle=|\lambda|^2||v||^2$, $||\lambda v||=|\lambda|\ ||v||$ #### Definition 6.10 -$v,u\in V$ are **orthogonal** if $=0$. +$v,u\in V$ are **orthogonal** if $\langle v,u\rangle=0$. #### Theorem 6.12 (Pythagorean Theorem) @@ -102,9 +102,9 @@ Proof: $$ \begin{aligned} - ||u+v||^2&=\\ - &=+\\ - &=+++\\ + ||u+v||^2&=\langle u+v,u+v\rangle\\ + &=\langle u,u+v\rangle+\langle v,u+v\rangle\\ + &=\langle u,u\rangle+\langle u,v\rangle+\langle v,u\rangle+\langle v,v\rangle\\ &=||u||^2+||v||^2 \end{aligned} $$