- #1
Bashyboy
- 1,421
- 5
Homework Statement
If ##A## is the subspace of ##l^\infty## consisting of all sequences of zeros and ones,
what is the induced metric on ##A##?
Homework Equations
The Attempt at a Solution
The metric imposed on ##l^\infty## is ##d(x,y) = \underset{i \in \mathbb{N}}{\sup} |x_i - y_i|##. I suspect that the induced (or, as a I call it, the reduced) metric is ##d(x,x) = 0## and ##d(x,y) = 1##. However, I am having difficulty showing this. Here is what I came up with:
Let ##I_{x,0} = \{i : x_i = 0\}## and ##I_{x,1} = \{i : x_i = 1\}##, and similarly define ##I_{y,0}## and ##I_{y,1}##. Hence, ##I_{x,1} \cap I_{y,1} \ne \emptyset## implies that there exists an ##i## such that ##x_i = 1## and ##y_ i = 1##; furthermore, ##x_i - y_i = 0##, which means ##\underset{i \in \mathbb{N}}{\sup} |x_i - y_i| = 0##...right?
This, however, does not appear to be a very elegant solution, as there will be many cases to deal with.