Differentiable Functions on Normed Linear Spaces

Differentiable Functions on Normed Linear Spaces In this article, we formalize differentiability of functions on normed linear spaces. Partial derivative, mean value theorem for vector-valued functions, continuous differentiability, etc. are formalized. As it is well known, there is no exact analog of the mean value theorem for vector-valued functions. However a certain type of generalization of the mean value theorem for vector-valued functions is obtained as follows: If ||ƒ'(x + t · h)|| is bounded for t between 0 and 1 by some constant M, then ||ƒ(x + t · h) - ƒ(x)|| ≤ M · ||h||. This theorem is called the mean value theorem for vector-valued functions. By this theorem, the relation between the (total) derivative and the partial derivatives of a function is derived [23].


Differentiable Functions on Normed Linear
Spaces 1 Yasunari Shidama Shinshu University Nagano, Japan Summary. In this article, we formalize differentiability of functions on normed linear spaces. Partial derivative, mean value theorem for vector-valued functions, continuous differentiability, etc. are formalized. As it is well known, there is no exact analog of the mean value theorem for vector-valued functions. However a certain type of generalization of the mean value theorem for vectorvalued functions is obtained as follows: If ||f (x + t · h)|| is bounded for t between 0 and 1 by some constant M, then ||f (x+t·h)−f (x)|| ≤ M ·||h||. This theorem is called the mean value theorem for vector-valued functions. By this theorem, the relation between the (total) derivative and the partial derivatives of a function is derived [23].

Preliminaries
In this paper r is a real number and S, T are non trivial real normed spaces. Next we state several propositions: (1) Let R be a function from R into S. Then R is rest-like if and only if for every real number r such that r > 0 there exists a real number d such that d > 0 and for every real number z such that z = 0 and |z| < d holds |z| −1 · R z < r.
(2) Let R be a rest of S. Suppose R 0 = 0 S . Let e be a real number. Suppose e > 0. Then there exists a real number d such that d > 0 and for every real number h such that |h| < d holds R h ≤ e · |h|. (3) For every rest R of S and for every bounded linear operator L from S into T holds L · R is a rest of T .
Let L 1 be a linear of S and L 2 be a bounded linear operator from S into T . Then Let x 0 be an element of R and g be a partial function from R to the carrier of S. Suppose g is differentiable in x 0 . Let f be a partial function from the carrier of S to the carrier of T . Suppose f is differentiable in ). (7) Let S be a real normed space, x 1 be a finite sequence of elements of S, and y 1 be a finite sequence of elements of R. Suppose len x 1 = len y 1 and for every element i of N such that i ∈ dom x 1 holds y 1 (i) = (x 1 ) i . Then x 1 ≤ y 1 . (8) Let S be a real normed space, x be a point of S, and N 1 , N 2 be neighbourhoods of x. Then N 1 ∩ N 2 is a neighbourhood of x. (9) For every non-empty finite sequence X and for every set x such that x ∈ X holds x is a finite sequence. Let G be a real norm space sequence. One can verify that G is constituted finite sequences.
Let G be a real linear space sequence, let z be an element of G, and let j be an element of dom G. Then z(j) is an element of G(j).
One can prove the following propositions: (10) The carrier of G = G. (11) Let i be an element of dom G, r be a set, and x be a function. If r ∈ the carrier of G(i) and x ∈ G, then x +· (i, r) ∈ the carrier of G. Let G be a real norm space sequence. We say that G is nontrivial if and only if: (Def. 1) For every element j of dom G holds G(j) is non trivial.
Let us mention that there exists a real norm space sequence which is nontrivial.
Let G be a nontrivial real norm space sequence and let i be an element of dom G. Note that G(i) is non trivial.
Let G be a nontrivial real norm space sequence. Note that G is non trivial.
The following propositions are true:  (12) Let G be a real norm space sequence, p, q be points of G, and r 0 , p 0 , q 0 be elements of G. Suppose p = p 0 and q = q 0 . Then p + q = r 0 if and only if for every element i of dom G holds r 0 (i) = p 0 (i) + q 0 (i).
(13) Let G be a real norm space sequence, p be a point of G, r be a real number, and r 0 , p 0 be elements of G. Suppose p = p 0 . Then r · p = r 0 if and only if for every element i of dom G holds r 0 (i) = r · p 0 (i).
(14) Let G be a real norm space sequence and p 0 be an element of G. Then 0 G = p 0 if and only if for every element i of dom G holds p 0 (i) = 0 G(i) .
(15) Let G be a real norm space sequence, p, q be points of G, and r 0 , p 0 , q 0 be elements of G. Suppose p = p 0 and q = q 0 . Then p − q = r 0 if and only if for every element i of dom G holds r 0 (i) = p 0 (i) − q 0 (i).

Mean Value Theorem for Vector-Valued Functions
Let S be a real linear space and let p, q be points of S. The functor ]p, q[ yielding a subset of S is defined as follows: Let S be a real linear space and let p, q be points of S. We introduce [p, q] as a synonym of L(p, q).
Next we state several propositions: (17) Let T be a non trivial real normed space and R be a partial function from R to T . Suppose R is total. Then R is rest-like if and only if for every real number r such that r > 0 there exists a real number d such that d > 0 and for every real number z such that z = 0 and |z| < d holds Rz |z| < r. (18) Let R be a function from R into R. Then R is rest-like if and only if for every real number r such that r > 0 there exists a real number d such that d > 0 and for every real number z such that z = 0 and |z| < d holds Let S, T be non trivial real normed spaces, f be a partial function from S to T , p, q be points of S, and M be a real number. Suppose that Then  (20) Let S, T be non trivial real normed spaces, f be a partial function from S to T , p, q be points of S, M be a real number, and L be a point of the real norm space of bounded linear operators from S into T . Suppose that Then

Partial Derivative of a Function of Several Variables
Let G be a real norm space sequence and let i be an element of dom G. The projection onto i yielding a function from G into G(i) is defined by: Let G be a real norm space sequence, let i be an element of dom G, and let x be an element of G. The functor reproj(i, x) yielding a function from G(i) into G is defined by: Let G be a nontrivial real norm space sequence and let j be a set. Let us assume that j ∈ dom G. The functor modetrans(G, j) yields an element of dom G and is defined by: Let G be a nontrivial real norm space sequence, let F be a non trivial real normed space, let i be a set, let f be a partial function from G to F , and let x be an element of G. We say that f is partially differentiable in x w.r.t. i if and only if: Let G be a nontrivial real norm space sequence, let F be a non trivial real normed space, let i be a set, let f be a partial function from G to F , and let x be a point of G. The functor partdiff(f, x, i) yielding a point of the real norm space of bounded linear operators from G(modetrans(G, i)) into F is defined as follows: ).

Linearity of Partial Differential Operator
For simplicity, we adopt the following rules: G denotes a nontrivial real norm space sequence, F denotes a non trivial real normed space, i denotes an element of dom G, f , f 1 , f 2 denote partial functions from G to F , x denotes a point of G, and X denotes a set.
Let G be a nontrivial real norm space sequence, let F be a non trivial real normed space, let i be a set, let f be a partial function from G to F , and let X be a set. We say that f is partially differentiable on X w.r.t. i if and only if: Next we state several propositions:  (25) For every set i such that i ∈ dom G and f is partially differentiable on X w.r.t. i holds X is a subset of G.
Let G be a nontrivial real norm space sequence, let S be a non trivial real normed space, and let i be a set. Let us assume that i ∈ dom G. Let f be a partial function from G to S and let X be a set. Let us assume that f is partially differentiable on X w.r.t. i. The functor f i X yields a partial function from G to the real norm space of bounded linear operators from G(modetrans(G, i)) into S and is defined by: (Def. 9) dom(f i X) = X and for every point x of G such that x ∈ X holds (f i X) x = partdiff(f, x, i).

Continuous Differentiatibility of Partial Derivative
Next we state the proposition Let G be a nontrivial real norm space sequence. One can verify that every point of G is len G-element.
We now state a number of propositions: (32) Let G be a nontrivial real norm space sequence, T be a non trivial real normed space, i be a set, Z be a subset of G, and f be a partial function from G to T . Suppose Z is open. Then f is partially differentiable on Z w.r.t. i if and only if Z ⊆ dom f and for every point x of G such that x ∈ Z holds f is partially differentiable in x w.r.t. i.
(35) Let x, y be points of G. Then (the projection onto i)(x + y) = (the projection onto i)(x) + (the projection onto i)(y).
(40) Let x be a point of G and a be an element of R. Then (the projection onto i)(a · x) = a · (the projection onto i)(x). (41) Let G be a nontrivial real norm space sequence, S be a non trivial real normed space, f be a partial function from G to S, x be a point of G, and i be a set.
(42) Let S be a real normed space and h, g be finite sequences of elements of S. Suppose len h = len g + 1 and for every natural number i such that Let G be a nontrivial real norm space sequence, x, y be elements of G, and Z be a set. Then x+·y Z is an element of G. (44) Let G be a nontrivial real norm space sequence, x, y be points of G, Z, x 0 be elements of G, and X be a set. If Z = 0 G and x 0 = x and y = Z+·x 0 X, then y ≤ x . (45) Let G be a nontrivial real norm space sequence, S be a non trivial real normed space, f be a partial function from G to S, and x, y be points of G. Then there exists a finite sequence h of elements of G and there exists a finite sequence g of elements of S and there exist elements Z, y 0 of G such that y 0 = y and Z = 0 G and len h = len G + 1 and len g = len G and for every natural number i such that i ∈ dom h holds h i = Z+·y 0 Seg((len G+1)− i) and for every natural number i such that i ∈ dom g holds (49) Let G be a nontrivial real norm space sequence, i, j be elements of dom G, x, y be points of G, and x 2 be a point of G(i). Suppose y = (reproj(i, x))(x 2 ) and i = j. Then (the projection onto j)(x) = (the projection onto j)(y).
(50) Let G be a nontrivial real norm space sequence, F be a non trivial real normed space, i be an element of dom G, x be a point of G, x 2 be a point of G(i), f be a partial function from G to F , and g be a partial function from G(i) to F . If (the projection onto i)(x) = x 2 and g = f · reproj(i, x), then g (x 2 ) = partdiff(f, x, i).
(51) Let G be a nontrivial real norm space sequence, F be a non trivial real normed space, f be a partial function from G to F , x be a point of G, i be a set, M be a real number, L be a point of the real norm space of bounded linear operators from G(modetrans(G, i)) into F , and p, q be points of G(modetrans(G, i)). Suppose that (52) Let G be a nontrivial real norm space sequence, x, y, z, w be points of G, i be an element of dom G, d be a real number, and p, q, r be points of G(i). Suppose y − x < d and z − x < d and p = (the projection onto i)(y) and z = (reproj(i, y))(q) and r ∈ [p, q] and w = (reproj(i, y))(r). Then w − x < d.
(53) Let G be a nontrivial real norm space sequence, S be a non trivial real normed space, f be a partial function from G to S, X be a subset of G, x, y, z be points of G, i be a set, p, q be points of G(modetrans(G, i)), and d, r be real numbers. Suppose that i ∈ dom G and X is open and x ∈ X and y−x < d and z−x < d and X ⊆ dom f and for every point x of G such that x ∈ X holds f is partially differentiable in x w.r.t. i and for every point z of G such that z −x < d holds z ∈ X and for every point z of G such that z − x < d holds partdiff(f, z, i) − partdiff(f, x, i) ≤ r and z = (reproj(modetrans(G, i), y))(p) and q = (the projection onto modetrans(G, i))(y).
(54) Let G be a nontrivial real norm space sequence, h be a finite sequence of elements of G, y, x be points of G, y 0 , Z be elements of G, and j be an element of N. Suppose y = y 0 and Z = 0 G and -10.2478/v10037-012-0005-1 Downloaded from PubFactory at 09/05/2016 08:39:48AM via Shinshu U Lib len h = len G + 1 and 1 ≤ j ≤ len G and for every natural number i such that i ∈ dom h holds h i = Z+·y 0 Seg((len G + 1) − i). Then x + h j = (reproj(modetrans(G, (len G + 1) − j), x + h j+1 ))((the projection onto modetrans(G, (len G + 1) − j))(x + y)). (55) Let G be a nontrivial real norm space sequence, h be a finite sequence of elements of G, y, x be points of G, y 0 , Z be elements of G, and j be an element of N. Suppose y = y 0 and Z = 0 G and len h = len G + 1 and 1 ≤ j ≤ len G and for every natural number i such that i ∈ dom h holds h i = Z+·y 0 Seg((len G + 1) − i). Then (the projection onto modetrans(G, (len G + 1) − j))(x + y) − (the projection onto modetrans(G, (len G + 1) − j))(x + h j+1 ) = (the projection onto modetrans(G, (len G + 1) − j))(y).
(56) Let G be a nontrivial real norm space sequence, S be a non trivial real normed space, f be a partial function from G to S, X be a subset of G, and x be a point of G. Suppose that (i) X is open, (ii) x ∈ X, and (iii) for every set i such that i ∈ dom G holds f is partially differentiable on X w.r.t. i and f i X is continuous on X.
Then (iv) f is differentiable in x, and (v) for every point h of G there exists a finite sequence w of elements of S such that dom w = dom G and for every set i such that i ∈ dom G holds w(i) = (partdiff(f, x, i))((the projection onto modetrans(G, i))(h)) and f (x)(h) = w. (57) Let G be a nontrivial real norm space sequence, F be a non trivial real normed space, f be a partial function from G to F , and X be a subset of G. Suppose X is open. Then for every set i such that i ∈ dom G holds f is partially differentiable on X w.r.t. i and f i X is continuous on X if and only if f is differentiable on X and f X is continuous on X.