A global genericity theorem for bifurcations in variational problems

Let C be a Banach space, H a Hilbert space, and let F(C,H) be the space of C^∞ functions f: C × H → R having Fredholm second derivative with respect to x at each (c, x) ?C × H for which $\text{D?}{}_{\mathrm{c}}\text{(x) = 0}$; here we write $\text{?}{}_{\mathrm{c}}\text{(x)}$ for $\text{?(c, x)}$. Say ? is of standard type if at all critical points of ?_c it is locally equivalent (as an unfolding) to a quadratic form Q plus an elementary catastrophe on the kernel of Q. It is proved that if f?F (A × B, H) satisfies a certain ‘general position’ condition, and dim B ? 5, then for most a?A the function f_o?F(B,H) is of standard type. Using this it is shown that those f?F(B,H) of standard type form an open dense set in F(B,H) with the Whitney topology. Thus both results are Hilbert-space versions of Thom's theorem for catastrophes in $\text{R}$ⁿ.