Metric Spaces

For 1≤p<∞, let us consider the following set of sequences in the linear space K, denoted by lp: lp = {(x1, x2, … ): xi ∈ K, $\sum _{i=1}^{\mathrm{\infty }}$ |xi|p<∞}. For x = (x1, x2, …) and y = (y1, y2, …) ∈ lp, define dp(x, y) = ($\sum _{i=1}^{\mathrm{\infty }}$ |xi – yi|p)1/p. Next, we prove that … Read more

A metric space is a set X equipped with a distance structure. Here we will study various properties of the metric space X that are well-behaved with respect to this distance structure. For example, the set ℝ of real numbers is a metric space with distance function d(x,y) = |x-y| for x, y ∈ ℝ. … Read more