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content/Math401/Extending_thesis/Math401_R1.md
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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: $\langle u,v\rangle=\overline{\langle v,u\rangle}$
<details>
<summary>Examples of inner product</summary>
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.
</details>
#### 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)
<details>
<summary>Examples of measure</summary>
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 $S$.
</details>
#### 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 $(X, v)$.
<details>
<summary>Examples of measurable space</summary>
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 $L^2(\mathbb{R}, \lambda)$.
<details>
#### 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 $\mathbb{R}^n$.

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#### 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.
<details>
<summary>Example of second countable space</summary>
Let $X=\mathbb{R}$ and $\mathcal{B}=\{(a,b)|a,b\in \mathbb{R},a<b\}$ (collection of all open intervals with rational endpoints).
Since the rational numbers are countable, so $\mathcal{B}$ is countable.
So $\mathbb{R}$ is second countable.
Likewise, $\mathbb{R}^n$ is also second countable.
</details>
<details>
<summary>Example of manifold</summary>
1-manifold is a curve and 2-manifold is a surface.
</details>
#### 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.