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Let $(M,g)$ be a compact Riemannian manifold without boundary. Let $V$ be the vector space $C^4(M) \times C^4(M)$, where addition and scalar multiplication are defined in the obvious way. Suppose we turn $V$ into an incomplete inner-product space with a inner product given by the second variation of a $C^2$ functional $E$ on $V$, $E=E(w,v)$, with respect to the second argument $v$. (It is symmetric and positive definite in this case.) Now let $X$ be a nowhere dense subset of $V$. Given this, can there exist a linear map $B$ from $X$ to $V$ that is one-to-one and onto?