Question

Q. If A is a subspace of l^∞ consisting of all sequence of 0 and 1. What is the induced metric on A?

Answer 1

Answer on Question #71759 – Math – Functional Analysis

Question

If $A$ is a subspace of $I^{\infty}$ consisting of all sequences of 0 and 1. What is the induced metric on $A$?

Solution

Recall that for any $x = (\xi_i) \in I^{\infty}$ and $y = (\eta_i) \in I^{\infty}$ we have that $d(x, y) = \sup_{i \in \mathbb{N}} |\xi_i - \eta_i|$. So, for any distinct $x, y \in A \subset I^{\infty}$, $d(x, y) = 1$ since they are sequences of zeros and ones. Thus, the induced metric on $A$ is the discrete metric, i.e. $d_A(x, y) = \begin{cases} 1, & x \neq y \\ 0, & x = y \end{cases}$.

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