Let $V$ be a finite dimensional vector space over a field $k$ with basis $B=\{v_i\}$. And let $V^*$ be its dual space with basis $B^*=\{\beta_i\}$, with $\beta_j(b_k)=\delta_{kj}$.
Show that $V^*$ is isomorphic with $V$.
We haven't been shown that two vector spaces are isomorphic if the have t...
Systolic geometry is a branch of differential geometry, a field within mathematics, studying problems such as the relationship between the area inside a closed curve C, and the length or perimeter of C. Since the area A may be small while the length l is large, when C looks elongated, the relationship can only take the form of an inequality. What is more, such an inequality would be an upper bound for A: there is no interesting lower bound just in terms of the length.
Mikhail Gromov once voiced the opinion that the isoperimetric inequality was known already to the Ancient Greeks. The mythological...
