Real Analysis

User Generated

abjurerobl

Mathematics

Description

In the pic

Unformatted Attachment Preview

Problem 1. (cf. P11.3) Let X be the set of all binary sequences and let d be the weighted Hamming distance. For very pe N and every d1,...,dp € {0,1} let Ud2.d = {r e X : (0) d; for i = 1,...,p}. (1) Prove that every Ud...,dp is an open set. (2) Prove that the collection U of all Ud...,dp is a basis for open sets in (X,d).
Purchase answer to see full attachment
User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.

Explanation & Answer

I wrote the solution, please read it and ask what is unclear.

For any 𝑝 ∈ ℕ and any 𝑑1 , 𝑑2 , … , 𝑑𝑝 ∈ {0, 1} denote
𝑈𝑑1 ,𝑑2 ,… ,𝑑𝑝 = {𝑥 ∈ 𝑋: 𝑥 𝑖 = 𝑑𝑖 , 𝑖 = 1, … , 𝑝}.
We need to prove that the sets of this form are open in the weighted Hamming distance
and that they form a basis for its open sets.
For the first we need to prove that any point of any such set has an open ball inside the set
(centered at that point). For the second it is enough to find a set of this form inside any
open ball (ask me why). So we need to analyze open balls in 𝑋 (with small radiuses
mostly).

1
The maximum distance is ∑∞
𝑖 =1 2𝑖 = 1. The difference between the first coordinates, if

nonzero (i.e. 1), contributes

1
2

to the distance an...


Anonymous
Goes above and beyond expectations!

Studypool
4.7
Trustpilot
4.5
Sitejabber
4.4

Related Tags