Notes - Metric Spaces MT23, Examples
Flashcards
Can you define the $d _ p$ metric?
\[d_p(v, w) = \left(\sum^n_{i=1} (v_i - w_i)^{p} \right)^{1/p}\]
Can you define the $l _ p$ norm?
\[||x||_p = \left(\sum^n_{i=1} x_i^{p} \right)^{1/p}\]
Can you define the $d _ \infty$ metric?
\[d\_\infty(v, w) = \max\_{i \in \\{1, \ldots, n\\}\,} |v\_i - w\_i|\]
Can you define the $l _ \infty$ norm?
\[||x||_\infty = \max_{i=1,\ldots,n} |x_i|\]
Can you define the $l _ p$ vector space?
\[\\{(x_n)_{n=1}^\infty : \sum ||x_n||_p < \infty\\}\]
Can you define the discrete metric?
\[d(x, y) = \begin{cases}1 &\text{if } x \ne y \\\\0 &\text{if } x = y\end{cases}\]
Can you define the $p$-adic metric?
\[d(x, y) = p^{-m}\]
where $m$ is the largest power of $p$ dividing $x - y$.
What property do some metrics (e.g. the $p$-adic metric) satisfy that is stronger than the triangle inequality?
\[d(x, z) \le \max(d(x, y), d(y, z))\]
Can you define the $L^1$ norm on $f \in C[0, 1]$?
\[||f||_1 = \int^1_0 |f(t)| \text dt\]