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Let $V$ be an n-dimensional real vector space, consider the space $F(V^p)$ of real functions on the p-fold cartesian product $V^p$ and its subspace $(V^{*})^p$ of multilinear functions (i.e. covariant tensors).

I know that every $f\in (V^*)^p$ can be written as a finite sum of pure tensors, i.e. (for simplicity I take $p=2$)

$f(v,w)= f_1(v)g_1(w) +\dots f_N(v) g_N(w)$

for some positive integer $N$ and $f_i,g_i\in V^*$. (In fact $N$ can be taken independent of $N$, the dimension of $(V^*)^p$ being finite).

Does a similar statement hold for $F(V^p)$ even if the latter is an infinite dimensional vector space? I.e. (in the case $p=2$ for simplicity) ), given $f\in F(V^2)$ do there exist a positive integer $N$ and functions $f_i,g_i:V\rightarrow \mathbb R$ for $i=1,\dots, N$ such that

$f(v,w)= f_1(v)g_1(w) +\dots f_N(v) g_N(w)$?

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    @Willie Wong. Yes, this is was I meant. And your counterexample gives an answer (which I would accept if you make it an official answer). Many thanks.2012-03-30

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Consider the following: let $f : V\times V\to \mathbb{R}$ be given by $f(v,w) = 0$ when $v\neq w$, and $f(v,w) = 1$ if $v = w$. (This is the generalisation of the Kronecker $\delta$ symbol, and can be also thought of as the characteristic function of the diagonal set in $V\times V$.) This function cannot be written as a finite (or even countably infinite) linear combination of functions of the form $g(v)h(w)$. (Since $V$ is a $\mathbb{R}$ vector space and contains uncountably many points.)

This shows that $F(V^2)$ cannot be decomposed as finite combinations of $F(V)$.