Showing that a functional can be fully described by vectors

Question

This is from a quantum mechanics textbook, so forgive me for my notation. I feel like I would get a better answer here so here I go. Say I have some vector space $V$ with basis $\left|1 \right\rangle, \left|2 \right\rangle,...,\left|n \right\rangle$. Now let $f$ be a functional on $V$. My textbook claims that

"$f$ is completely determined by the $n$ values $f(\left|1 > \right\rangle),...,f(\left|n \right\rangle)$".

First off, what does it mean to be 'completely determined'? How can I show that this claim is true?

Answer 1

A functional in this context is a linear map.

By definition of basis, every quantum state can be written as

$$| \phi \rangle=\sum_{i=1}^nc_i|i\rangle$$

Hence

$$f(| \phi \rangle)=f\left(\sum_{i=1}^nc_i|i\rangle\right)=\sum_{i=1}^nc_if\left(|i\rangle\right)$$

Hence knowing the value of $f(|i\rangle$) enables us to evaluate $f(|\phi\rangle)$ if we know the $c_i$.