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2. Linear Transformations
Let X , Y be Banach spaces. A mapping f : X → Y is linear if it satisfies the following two properties:
- f (x + y) = f (x) + f (y) for all x, y ∈ X
- f (αx) = αf (x) for all x ∈ X , α ∈ R.
A linear map f : X → Y is called bounded if there is a constant C > 0 such that | f (x) | ≤ C| x | for all x ∈ X. Facts:
- A linear map f is bounded if and only if it is continuous.
- The linear map f is bounded if and only if the quantity sup| x |≤ 1 | f (x) | is finite.
- The quantity sup| x |≤ 1 | f (x) | in the preceding statement is also equal to sup| x |=1 | f (x) |
- Every linear map whose domain is Rn^ or Cn^ is bounded (hence con- tinuous).
If f is a bounded linear map (transformation), we set | f | = sup| x |=1 | f (x) |.
This defines a norm in the space L(X , Y) of bounded linear maps from X to Y, making it into a Banach space also.
Fixed Point Theorems
Many existence theorems for differential equations can be reduced to fixed point theorems in appropriate function spaces. Here we will discuss a few relevant results. Let X be a metric space and let T : X → X be a mapping. A fixed point of T is a point x ∈ X such that T (x) = x. A self-map T of a metric space X is called a contraction (or contraction map or mapping) if there is a constant 0 < λ < 1 such that
d(T x, T y) ≤ λd(x, y)
for all x, y ∈ X. Thus, T : X → X is a contraction if and only it is Lipschitz with Lipschitz constant less than 1. Theorem. (Contraction Mapping Theorem) Suppose X is a complete metric space and T : X → X is a contraction map. Then, T has a unique fixed point x¯ in X. Moreover, if x is any point in F, then the sequence of iterates x, T x, T 2 x,... converges to x¯ exponentially fast. Proof. Uniqueness: If 0 < λ < 1 is the contraction constant for T and T x = x, T y = y, then
d(x, y) = d(T x, T y) ≤ λd(x, y)
which implies that d(x, y) = 0. This in turn implies that x = y. QED. Existence: Let x 0 = x, x 1 = T x, xi = T ix,.. .. Then, d(xn+1, xn) ≤ λd(xn, xn− 1 ) ≤... ≤ λnd(x 1 , x 0 ) for 1 ≤ n. Thus, for m > n,
d(xm, xn) ≤ d(xm, xm− 1 ) + d(xm− 1 , xm− 2 ) +... + d(xn+1, xn) ≤ (λm−^1 + λm−^2 +... + λn)d(x 1 , x 0 )
=
λn(1 − λm−n) 1 − λ
d(x 1 , x 0 ) ≤ Cλnd(x 1 , x 0 )
This implies that the sequence x 1 , x 2 ,... is a Cauchy sequence. By com- pleteness of X, it converges, say to an element ¯x of X. But, since T is continuous,
T (¯x) = T (limn→∞xn) = limn→∞T (xn) = limn→∞ xn+1 = ¯x,
so, T (¯x) = ¯x. This proves the existence and the exponential convergence. QED.
There is another useful criterion for the existence of fixed points of trans- formations in Banach spaces. Let X be a Banach space. Let x, y ∈ X. The line segment in X from x to y is the set of points {(1 − t)x + ty : 0 ≤ t ≤ 1 }. A subset F of X is called convex if for any two points x, y ∈ F , each point in the line segment from x to y is contained in F. Examples:
- Linear subspaces are convex.
- open and closed balls are convex
The following are three remarkable theorems. Theorem.(Brouwer Fixed Point Theorem). Every continuous map T of the closed unit ball in Rn^ to itself has a fixed point. Theorem.(Schauder Fixed Point Theorem). Every continuous self-map of a compact convex subset of a Banach space has a fixed point. Theorem.(Schauder-Tychonov Fixed Point Theorem).Every continuous self-map of a compact convex subset of a locally convex linear topological space to itself has a fixed point.
