Metric Spaces MT23, Examples
Flashcards
Can you define the $d _ p$ metric?
\[d _ p(v, w) = \left(\sum^n _ {i=1} \vert v _ i - w _ i \vert ^{p} \right)^{1/p}\]
Can you define the $l _ p$ norm?
\[ \vert \vert x \vert \vert _ p = \left(\sum^n _ {i=1} \vert x _ i \vert ^{p} \right)^{1/p}\]
Can you define the $d _ \infty$ metric?
\[d _ \infty(v, w) = \max _ {i \in \{1, \ldots, n\}\,} \vert v _ i - w _ i \vert \]
Can you define the $l _ \infty$ norm?
\[ \vert \vert x \vert \vert _ \infty = \max _ {i=1,\ldots,n} \vert x _ i \vert \]
Can you define the $l _ p$ vector space?
\[\{(x _ n) _ {n=1}^\infty : \sum \vert x _ n \vert ^p < \infty\}\]
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 $f \in C[0, 1]$?
\[ \vert \vert f \vert \vert _ 1 = \int^1 _ 0 \vert f(t) \vert \mathrm dt\]