Apologies if this is a really trivial question, but I've never been quite sure as to the usage of the terminology(adsbygoogle = window.adsbygoogle || []).push({}); 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 itdualto ##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 bydualin that if we no the form of one then we can determine the other?

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# What is the meaning of the term "dual"?

Loading...

Similar Threads - meaning term dual | Date |
---|---|

A The meaning of the commutator for two operators | Jan 9, 2018 |

I Meaning of mapping R[X]->Maps[R,R] | Apr 15, 2017 |

I Express power sums in terms of elementary symmetric function | Feb 22, 2017 |

I What does this symbol mean?? | Oct 27, 2016 |

I Matrix Rings - Basic Problem with Meaning of Notation | Oct 8, 2016 |

**Physics Forums - The Fusion of Science and Community**