A problem about dual basis I've got a problem: 
Let $V$ be the vector space of all functions from a set $S$ to a field $F$:
$(f+g)(x) = f(x) + g(x)\\
(\lambda f)(x) = \lambda f(x)$
Let $W$ be any $n$-dimensional subspace of $V$. Show that there exist points $x_1, x_2, \dots, x_n$ in $S$ and funtions $f_1, f_2, \dots, f_n$ in $W$ such that $f_i(x_j) = \delta_{ij}$
I'm confusing about set $S$. In the case $S$ has less than $n$ elements, how can I chose $x_1, x_2, \dots, x_n$? Anyone can help me solve this problem?
thank you.
 A: Assuming $S$ is a finite set, the dimension of the $F$-vector space $V$ of functions $S \to F$ is equal to the size of $S$. Therefore if $V$ has an $n$-dimensional subspace, then $\dim(V) \ge n$, so $S$ must have at least $n$ elements. This means that the problematic case that you are considering can't arise.
A: If $S$ is finite, say $S = \{\xi_1, \ldots, \xi_k\}$, then the map
$$ V \to K^k, \quad f \mapsto \bigl(f(\xi_1), \ldots, f(\xi_k)\bigr) $$
is an isomorphism. So $V$ has a $n$-dimensional subspace iff $\lvert S\rvert \ge n$.
A: This is not a dual basis question. The dual of a vector space $V$ over field $k$ is the set of all linear functions from $V$ into $k$.
In your case the functions do not have to be linear, or even continuous. An example of what you have is $\mathbb R^n$, which can be thought of as the set of all functions from the set $\lbrace  1,2, \ldots n\rbrace$ into the field $\mathbb R$. These functions take the number $i$ and return the vector's $i^{th}$ component.  We just usually write the vectors in the form $ \left( \begin{array}{ccc}
f(1) \\
f(2) \\
... \\
f(n) \end{array} \right)$. Now imagine we are actually working in the vector space $\mathbb R^n$ and that it contains an $m$ dimensional subspace. What does this tell us about the size of $n$? And what kind of vectors exist in a space this large?
The dual to the space $\mathbb R^n$ of column vectors, is the space of length $n$ row vectors. These work like functions on column vectors through matrix multiplication. The row vectors are functions $v: \lbrace 1,2, \ldots, n \rbrace \to \mathbb R$. The dual basis elements are (linear) functions $\alpha: \mathbb R^n \to \mathbb R$
