Functional Derivatives and Differentiability in Density-Functional Theory
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Definition 5 Let T be an operator (mapping, transformation) whose domain Dom(T)
and range Ran(T) belong to metric spaces (X, d X ) and (Y, d Y ), respectively. The operator T is continuous at point x 0 ∈ Dom(T) if, for every 𝜖 > 0, there exists 𝛿 > 0 such
that
d Y (Tx, Tx 0 ) < 𝜖
(102)
whenever
d X (x, x 0 ) < 𝛿 .
(103)
Definition 6 A sequence {x (k) } in a metric space (S, d) is said to be a Cauchy
sequence if d(x
(k)
, x
(l)
) → 0 as k, l → ∞. This means that for every 𝛿 > 0 there exists
N 𝛿 such that d(x (k) , x (l) ) ≤ 𝛿 for any k, l ≥ N 𝛿 .
Definition 7 A metric space (S, d) is said to be complete if every Cauchy sequence
in (S, d) has a limit in (S, d).
Definition 8 A norm (or length function) on a vector space  is a real-valued function, ||x||, defined for all vectors x ∈  and which satisfies the following axioms:
1. ||x|| > 0; ||x|| = 0 if and only if x = 𝜃;
2. ||x + y|| ≤ ||x|| + ||y||, ∀x, y ∈ ;
3. ||𝜆x|| = |𝜆| ⋅ ||x||, for an arbitrary scalar 𝜆.
A normed vector space, denoted by (, || ⋅ ||) consists of a vector space  and a
norm || ⋅ || on .
Definition 9 A complete (with respect to the norm) normed vector space is called
a Banach space.
Definition 10 Let T ∶  →  be a bounded linear transformation, that is,
||Tx|| ≤ K||x||.
(104)
The smallest value of K which satisfies this inequality is denoted by ||T|| and called
the norm of T. It can be verified that this norm for operators satisfies the axioms for a
norm function and that we may therefore talk of the vector space of bounded linear
transformations T ∶  →  . This normed vector space is denoted by (,  ).
Definition 11 Consider an operator T ∶  →  where  is a vector space and 
is a normed vector space. Let the domain of the operator T, Dom(T) ⊂ , and s ∈ :
if the limit
dT(x; s) = lim
𝜆→0
T(x + 𝜆s) − T(x)
𝜆
(105)
exists, it is called the Gâteaux differential of T at x in the direction s. The limit is
to be understood in the sense of convergence with respect to the norm in  . The
differential may exist for some s and fail to exist for others: if the differential exists
at x for all s we say that T is Gâteaux differentiable at x.