Mathematics Prelims

July 14, 2008

The Sequence Space l^p

Filed under: Analysis,Functional Analysis — cjohnson @ 7:16 pm

If p \geq 1, then the space l^p represents the set of all sequences (x_n) where the following converges for each (x_n) \in l^p.

\sum_{n = 1}^\infty |x_n|^p < \infty.

The metric for the l^p spaces is given below, where x = (x_n) and y = (y_n).

d(x, y) = (\sum_{n=1}^\infty |x_n - y_n|^p)^{1/p}

To show this is indeed a metric we must show the four properties of a metric are satisfied.  We see that d(x, y) >= 0 since we have a sum of non-negative terms.  The distance between x and y is finite due to Minkowski.  If d(x, y) = 0, we must have that each term in our sum is zero (since they’re all non-negative), meaning x_n = y_n for each n, so x = y.  If x = y, then we’ll have x_n - y_n = 0 for each n, so d(x, y) = 0.  Symmetry is due to the fact that |a - b| = |(-1) (b - a)| = |-1| |b - a| = |b - a|.  Finally, the triangle inequality is pretty much just a restatement of Minkowski, but where we add and subtract by a z_n term in the absolute value on the left-hand side.  That is, suppose z = (z_n) and that |x_n - y_n| = |x_n - z_n + z_n - y_n| and apply Minkowski.

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