Triangle Inequality
\[|A+B| \leq |A| + |B|\]
\[\left|\int_0^tf(t)g(t)dt\right| \leq \int_0^t|f(t)g(t)|dt\]
proof:
\[-|A|\leq A \leq |A|\]
\[-|B| \leq B \leq |B| \]
\[-|A|-|B| \leq A+B \leq |A|+|B|\]
\[|A+B| \leq |A|+|B|\]
Cauchy-Schwarz Inequality
\[|\langle\textbf{v},\textbf{w}\rangle|^2\leq\langle\textbf{v},\textbf{v} \rangle\cdot\langle\textbf{w},\textbf{w} \rangle\]
\[\left|\int_0^tf(t)g(t)dt\right|^2 \leq \left|\int_0^tf^2(t)dt \right| \left|\int_0^tg^2(t)dt \right|\]
Minkowski Ineqautliy
\[||f+g||_p \leq ||f||_p+||g||_p\]
\[\left(\sum_{k=1}^n|x_k+y_k|^p\right)^{\frac{1}{p}} \leq \left( \sum_{k=1}^n |x_k|^p\right)^{\frac{1}{p}} + \left( \sum_{k=1}^n |y_k|^p\right)^{\frac{1}{p}}\]
\(p=2\)일 때
\[\sqrt{\sum_{k=1}^n|x_k+y_k|^2} \leq \sqrt{ \sum_{k=1}^n |x_k|^2} + \sqrt{\sum_{k=1}^n |y_k|^2}\]
복소수의 triangle inequality를 증명할 때 사용됨.
Jensen's Inequality
Let \(\varphi\) be a convex function and \(X\) be a random variable.
\[\varphi(\mathbb{E}(X)) \leq \mathbb{E}(\varphi(X))\]
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