Instructor: Prof. Krishna Hanumanthu Link to lectures ROUGH NOTES (!) Lec-1: Rings (Commutative rings with unity. So no to matrix rings). Eg : $\mathbb{Z}, \mathbb{Q}, \mathbb{Z}[i] := \lbrace x + iy : x,y \in \mathbb{Z} \rbrace$ are rings. $\frac{1}{2} \mathbb{Z}$ isn’t, as it has $\frac{1}{2}$ but not $\frac{1}{2} \cdot \frac{1}{2} = \frac{... Read more 25 Sep 2021 - 18 minute read
Consider an integrable decreasing function $ f : [2, \infty) \to \mathbb{R}_{> 0} $. $ \int_{2}^{x} f(t) dt = \int_{2}^{3} f + \ldots + \int_{\lfloor x \rfloor -1}^{\lfloor x \rfloor} f + \int_{\lfloor x \rfloor}^{x} f $. Using $ f(i+1) \leq \int_{i}^{i+1} f \leq f(i) $, we get $ f(3) + \ldots + f(\lfloor x \rfloor) + \int_{\lfloor x \rfloo... Read more 08 Aug 2021 - 1 minute read
[$\color{goldenrod}{\underline{\text{This}}}$ is the first part.] Here is the “right version”. For a change we’ll take $ \mathcal{C}([a,b]) $ instead of $ \mathbb{R}^n $ (The proof structure is esentially the same). Take $ \langle f, g \rangle := \int_{a}^{b} f(t) g(t) dt $. Also for real $ p \geq 1 $, $ \lVert f \rVert_{p} := \left( \int \vert... Read more 08 Aug 2021 - 2 minute read
$ \log(n!) = \log(1) + \ldots + \log(n) $. This sum is $ \int_{1}^{2} \log(2) + \ldots + \int_{n-1}^{n} \log(n) $ $ \geq \int_{1}^{2} \log(t)dt + \ldots + \int_{n-1}^{n} \log(t)dt $ $ = \int_{1}^{n} \log(t) dt = n \log(n) - n +1 $. Also $ \log(2) + \ldots + \log(n-1) $ $ = \int_{2}^{3} \log(2) + \ldots + \int_{n-1}^{n} \log(n-1) $ $ \leq \int_... Read more 07 Aug 2021 - less than 1 minute read
[This is a weaker version, but the proof comes naturally] Consider $ \mathbb{R}^n $. We know $\lVert x\rVert_2 := \sqrt{\sum x^2_i} $ is a norm (ie “behaves like length”, especially $ \lVert x+y\rVert_2 \leq \lVert x\rVert_2 + \lVert y\rVert_2 $). So we can ask ourselves : $ \color{goldenrod}{\text{Q}} $) For integer $ p\geq 1$, taking $ \lVer... Read more 07 Aug 2021 - 3 minute read