- #1

- 601

- 7

## Main Question or Discussion Point

Apologies if this is a really trivial question, but I've never been quite sure as to the usage of the terminology

*dual space*. I get that given a vector space ##V## we can construct a set of linear functionals that map ##V## into its underlying field and that these linear functionals themselves form a vector space ##V^{\ast}##, but what is meant by calling it*dual*to ##V##? Is it simply that given one we can construct the other and so they are intricately related to one another? For example, given a vector ##\mathbf{v}\in V##, and a basis ##\mathcal{B}=\lbrace\mathbf{e}_{i}\rbrace##, such that ##\mathbf{v}=a^{i}\mathbf{e}_{i}##, then we can define a unique map ##f\in V^{\ast}## such that ##f(\mathbf{v})=a^{i}##. As this map is linear it follows that $$f(\mathbf{v})=a^{i}f(e_{i})$$ and so ##f## is determined uniquely by its action on the basis vectors ##\lbrace\mathbf{e}_{i}\rbrace##. Is this what is meant by*dual*in that if we no the form of one then we can determine the other?