If , then the space represents the set of all sequences where the following converges for each .

.

The metric for the spaces is given below, where and .

To show this is indeed a metric we must show the four properties of a metric are satisfied. We see that since we have a sum of non-negative terms. The distance between x and y is finite due to Minkowski. If , we must have that each term in our sum is zero (since they’re all non-negative), meaning for each , so . If , then we’ll have for each , so . Symmetry is due to the fact that . Finally, the triangle inequality is pretty much just a restatement of Minkowski, but where we add and subtract by a term in the absolute value on the left-hand side. That is, suppose and that and apply Minkowski.

### Like this:

Like Loading...

*Related*

## Leave a Reply