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P. Xiang and Y. A. Wang
Appendix 1
Here, we will briefly introduce some mathematical concepts relevant to our discussion in the main text. All the following content are adopted from an introductory
book on functional analysis [26].
Definition 1 A vector space  is a set of elements called vectors with two operations called addition and scalar multiplication, which satisfy the following axioms.
∙ Addition axioms: To every pair of vectors x, y ∈ , there corresponds a unique
vector x + y ∈ , the sum of x and y, such that
1. x + y = y + x;
2. (x + y) + z = x + (y + z);
3. there exists a unique zero vector 𝜃 ∈  such that x + 𝜃 = 𝜃 + x = x, ∀x ∈ ;
4. for every vector x there exists a unique vector (−x) ∈  such that x + (−x) = 𝜃.
∙ Scalar multiplication axioms: To every scalar 𝛼 and every vector x ∈  there corresponds a unique vector 𝛼x ∈  such that
1. 𝛼(𝛽x) = (𝛼𝛽)x for every scalar 𝛽;
2. 1x = x, 0x = 0, ∀x ∈ ;
3. 𝛼(x + y) = 𝛼x + 𝛼y and (𝛼 + 𝛽)x = 𝛼x + 𝛽x.
Definition 2 If x and y are two points of a vector space, then the line segment joining
them is the set of elements {𝛽x + (1 − 𝛽)y | 0 ≤ 𝛽 ≤ 1}. A subset S of a vector space
is convex if the line segment of joining any two points in S is contained in S.
Definition 3 Let  and  be two vector spaces with the same system of scalars.
Then a function (or mapping) that maps uniquely the elements of  onto elements
of  ,
T ∶  → 
(101)
is called a linear transformation of  into  if
1. T(x + y) = Tx + Ty, ∀x, y ∈ ;
2. T(𝛼x) = 𝛼Tx, ∀x ∈  and for all scalars 𝛼.
Definition 4 A metric (or distance function) on a set S is a real-valued function
d(x, y) defined for all pairs of elements x and y in S and which satisfies the following
axioms:
1. d(x, y) > 0; d(x, y) = 0, if and only if x = y;
2. d(x, y) = d(y, x), ∀x, y ∈ S;
3. d(x, z) ≤ d(x, y) + d(y, z), ∀x, y, z ∈ S.
A metric space denoted by (S, d) consists of a set S and a metric d on S.