By Lee J.M.

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K}. , vk ) for all permutations σ. The set of all bounded symmetric (resp. skew-symmetric) multilinear maps V × · · · × V → W is denoted Lksym (V; W) (resp. Lkskew (V; W) or Lkalt (V; W)). Now if the W is complete, that is, if W is a Banach space then the space L(V, W) is a Banach space in its own right with norm given by A = sup v∈V,v=0 A(v) W = sup{ A(v) v V W : v V = 1}. , Ek ; W) is continuous if and only if it is bounded. We shall need to have several normed spaces handy for examples. 16 CHAPTER 1.

E × F → R be a fixed E−nondegenerate pairing. If F : U ⊂ F → R is differentiable at f ∈ F then the unique element δF δf ∈ E such δF that δf , . = DF |f is called the functional derivative of F . In the special (but common) case that both E and F are spaces of functions defined on a common measure space and if the pairing is given by integration h, g := h(x)g(x)µ(dx) with respect to some measure µ, then we sometimes δF write δfδF (x) instead of δf (x). The only reason for this is to harmonize with existing notation used in the naive versions of the functional derivative found in the physics literature.

Let GL(n, R) be the nonsingular n × n matrices and show that GL(n, R) is an open subset of the vector space of all matrices Mn×n (R) and then find the derivative of the determinant map: det : GL(n, R) → R (for each A this should end up being a linear map D det|A : Mn×n (R) → R). What is ∂ ∂xij det X where X = (xij ) ? 7. Let A : U ⊂ E → L(F, F) be a C r map and define F : U × F → F by F (u, f ) := A(u)f . Show that F is also C r . Hint: Leibniz rule theorem. 8. Show that if F is any closed subset of Rn there is a C ∞ -function f whose zero set {x : f (x) = 0} is exactly F .