Master Dissertation

Theorem 1.2. Let d : R N × R N → R be defined by
d((an), (bn)) = 2 −k , where k ∈ N0
is the smallest index such that ak = bk. If no such k exists we define
d((an), (bn)) = 0. Then (R N , d) is a metric space.
Proof. The only non-trivial part is the triangle inequality.
Let d(a, b) = 2−k1 −k2 , d(b, c) = 2
exist.
−k3 and d(a, c) = 2 assuming k1, k2 and k3
Say, k3 ≥ k1, then 2−k3 −k1 −k1 −k2 ≤ 2 ≤ 2 + 2 , hence we may assume
= bk3 and bk3 = ck3 thus ak3 = ck3 ,
k3 < ki for i = 1, 2. Then ak3
contradicting the definition of k3.
If k3 doesn’t exist the inequality is obvious. Say, k1 doesn’t exist, then
(an) = (bn) hence k2 = k3 if they exist. If not, then (an) = (cn).
Theorem 1.3. The operations of addition and multiplication are
continuous.
Proof. Say, fn → f and gn → g then we need to prove that
fn + gn → f + g, that is d(fn + gn, f + g) → 0 for n → ∞.
We need to prove that given ɛ > 0 there exists a δ such that
max{d(fn, f), d(gn, g)} < δ ⇒ d(fn + gn, f + g) < ɛ.
Given ɛ > 0 find k such that 2 −k < ɛ. Let δ = 2 −k /2 = 2 −(k+1) , then
|(f − fn)h| + |(g − gn)h| = 0, for all h ≤ k + 1. (1.1)
Hence since 2 −(k+1) + 2 −(k+1) = 2 −k
d(fn + gn, f + g) < 2 −k < ɛ.
Note that equation (1.1) also implies |(fngn)h| − |(fg)h|=0 for h ≤ k + 1,
hence multiplication is also continuous.
The cases where no k exists are obvious.
This leads to the following result.
Corollary 1.4. (RN , d) is a topological ring and
(an) =
anX n .
n≥0
Note that in this ring convergence is absolute, in fact any rearrangements
of the series converges to the same limit.
Definition 1.5. We denote the topological ring (R N , d) by R[[X]] and call
it the ring of formal power series.
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