# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 23: Dual Spaces

A linear functional is a mapping $f: V \rightarrow F$ from a vector space $V$ to its field $F$.

$V ^*$, the dual space of $V$, is the vector space of all such linear functionals, with addition of functionals and multiplication by scalars from $F$ being defined in the expected manner.

If $V$ is finite-dimensional, then $\textrm{dim}(V ^*) = \textrm{dim}(V)$ and $V ^*$ is isomorphic to $V$, but this is not the case when $V$ is infinite-dimensional.