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.