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Random math: 23

Source of most content: Gunhee Cho and Lecture note by Matsumura Closed set Def. Given T.S. $(X, T_X)$, $A \subseteq X$ is closed if $A^c$ is open. NB1; Let $X = [ 0,1 ] \cup (2,3) \subseteq (\mathbb{R}, T_{std})$. Consider $(X, T_{subsp})$. Then, $(2,3) = X \setminus [0 ,1]: (2, 3)$ is complement of open set, i.e., $(2,3)$ is closed ...

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Random math: 22

Source of most content: Gunhee Cho and Lecture note by Matsumura Topology on $\mathbb{R}$ Def. Given a set $X=\mathbb{R}$, we call a topology $T_{std}$ generated by basis $\mathcal{B} = \{ (a,b) : a < b, a,b \in \mathbb{R} \}$ the standard topology on $\mathbb{R}$. Def. Given a set $X=\mathbb{R}$, we call a topology $T_{l}$ generated by ...

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Random math: 21

Source of most content: Gunhee Cho and Lecture note by Matsumura Basis and Topology Def. Given a set $X \ne \emptyset$, we call $\mathcal{B} (\subseteq 2^X)$ the basis on $X$ if it satisifes: $\mathcal{B}$ covers $X$ $\Leftrightarrow \forall x \in X, \exists B \in \mathcal{B}$ s.t. $x \in B$ $\Leftrightarrow X = \cup_{B \in \mathcal{B}...

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Random math: 20

Source of most content: Gunhee Cho SVD \[\text{Let } A \in Mat_{m\times n}(F = \mathbb{R} \text{ or } \mathbb{C}). \text{ Then, } A \text{ can be decomposed as of the form } A = U \Sigma V^H. \nonumber\] Here, $U$ is an $m\times m$ unitary/orthogonal matrix, $\Sigma$ is an $m\times n$ diagonal matrix with non-negative real number, and $V$ is ...

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Random math: 19

Source of most content: Gunhee Cho Diagonalization Recall that the orthonormal basis(ONB) consisting of eigenvectors of $T$ exists iff $T$ is normal as shown in RM 18. Corollary \[A \in Mat_{n \times n} (\mathbb{C}) \text{ is normal iff } \exists \text{ unitary } Q \text{ s.t. } Q^*AQ \text{ is a diagonal matrix.} \nonumber\] Proof Since $...

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Random math: 18

Source of most content: Gunhee Cho Schur’s theorem \[\text{Given a finite dimensional inner product space (f.d.i.p.s) } (V, \langle , \rangle) \text{ over } F,\ \text{let linear operator } T: V\rightarrow V. \nonumber\] \[\text{Assume } \varphi_T (t) (:= \det (T-t Id)) \text{ splits. Then, there exists orthonormal basis } \beta \text{ for } V...

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Random math: 17

Source of most content: Gunhee Cho Dual space Def. Given a finite dimensional vector space $V$ over $F$, Define $V^* := \mathcal{L}(V, F) = \{ f: V \rightarrow F \mid f \text{ is linear} \}$, which is called the dual space of $V$. NB1; $ V^* $ is a VS over $F$ when we equipped with $\forall f,g \in V^*,\ (f+g)(v) := f(v) + g(v)), \forall c \i...

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Random math: 16

Source of most content: Gunhee Cho Inner product space Def. Given a finite dimenstional vector space $V$ over $F$, we call a function $\langle\ , \rangle: V \times V \rightarrow F$ the inner product if it satisfies: (Linearity w.r.t. 1st component) $\forall x,y,z \in V, \langle x+y, z\rangle = \langle x+z\rangle + \langle x, z\rangle.$...

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