## 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 ...

## 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 ...

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}... Read more ## 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 ... Read more ## 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$...

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...