TITLE: What is a dual space?
QUESTION [5 upvotes]: I've started studying differential geometry by myself and I ran into dual spaces in a section on 1-forms. I'm not very well versed in linear algebra so any help is much appreciated.

REPLY [6 votes]: In the context of vector spaces, the dual space is a space of linear "measurements". When a dual vector $f$ acts on a vector $v$, the scalar output $f(v)$ provides information about $v$; in particular, it gives you something about a coordinate of $v$ in some direction. A vector $v$ can be reconstructed from knowing all the values $f(v)$ for $f$ in the dual space.
As a rough example from engineering, consider vector space $V$ the space of continuous, periodic functions on $[0, 2\pi]$. Then there are certain dual vectors $f_k$ which returns how much of the function $v(x) \in V$ contains a frequency $k$. So if $v(x)$ was a sound wave, then $f_k(v)$ tells you whether the musical note of frequency $k$ is in the sound. Then the entire sound wave $v(x)$ can be reconstructed from knowing $f_k(v)$ for all $k$, since that would tell you all the notes in the sound.
And of course, $f_k(v)$ is going to be the $k$th Fourier coefficient of $v$.