[Ref: Prof S. G. Johnson’s proof] Let V be an R−vector space with basis B=(v1,…,vn), and let ‖ be two norms on it. Turns out these norms are equivalent. If { \mathscr{V} \overset{T}{\to} \mathscr{W} } is an isomorphism of ${\mathbb{R}-}... Read more 26 Nov 2021 - 3 minute read
Take an interval of the form { [\delta, 2A-\delta] \subseteq \mathbb{R} _{\gt 0} }. So here { 0 \lt \delta \lt A }. Consider { \log (x) } on { [\delta, 2A-\delta] }. We’ll find a sequence { (P _n (x)) } of polynomials of degrees { \deg(P _n) \leq n } such that { \max _{x \in [\delta, 2A - \delta]} \vert \log(x) - P _n (x) \vert }... Read more 22 Nov 2021 - 2 minute read
[Minor observation] Notation: Let { (x _j) _{j \geq 1} } be a sequence in a normed space. A sequence { (x _j) _{ j \in J} } with { J \subseteq \mathbb{Z} _{\gt 0} } infinite is called a subsequence. We say subsequence { (x _j) _{j \in J} } converges to { p } as { j \in J, j \to \infty } if for every { \epsilon \gt 0 }, ${ x _j ... Read more 11 Nov 2021 - 3 minute read
[An old proof. “By finding scaled versions of the triangle within itself”. Yet somewhat mysterious.] So {c = ( \frac{b}{c} ) b + ( \frac{a}{c} ) a. } Read more 09 Oct 2021 - less than 1 minute read
Instructor: Prof. A. V. Jayanthan Link to lectures Book: Atiyah-Macdonald ROUGH NOTES Lec-1: Rings, subrings, homomorphisms, ideals Lec-2: Prime and maximal ideals Lec-3: Every nonzero ring has a maximal ideal. (Apply this to A/I where ideal I \subsetneq A. From correspondence thm, there is a maximal ideal containing I. Especially ev... Read more 06 Oct 2021 - 20 minute read