Infinite sum of squares converges

In summary, the conversation discusses proving that L2, the set of all infinite sequences of real numbers whose sum of squares converges, is a metric. The first step is to show that if xi and yi are in L2, then xi-yi is also in L2. This is done by breaking up the squared quantity and showing that the sum of xi*yi is also finite. The conversation then moves on to discussing the remaining parts of the definition of a metric, with the use of the Cauchy-Schwarz inequality to prove them.
  • #1
rjw5002

Homework Statement


So, L2 is defined to be the set of all infinite sequences of real numbers, s.t. the sum of their squares converges:
L2 = {x=(x1,...,xn,...) | [tex]\Sigma[/tex]xi < [tex]\infty[/tex]}
we have d(x,y) = [tex]\sqrt{\Sigma (xi-yi)^2}[/tex]

I need to show that this is a metric, starting by showing that if [tex]xi,yi\in[/tex] then [tex]xi-yi\in[/tex]

Homework Equations



definition of a metric

The Attempt at a Solution


so my initial problem was with the first step:
x = (x1,..., xn,...), y = (y1,...,yn,...), and then x - y = (x1 - y1, ...,xn-yn,...)
to show [tex]xi,yi\in[/tex], we must show that [tex]\Sigma (xi -yi) < \infty[/tex].
First, I broke up the squared quantity to get [tex]\Sigma (xi^2- 2xi*yi + yi^2) [/tex].
Carrying the sumation through, [tex]\Sigma xi^2 < \infty [/tex] (as for yi^2).
But where I get stuck is how to show or determine that [tex]\Sigma xi*yi < \infty [/tex].

So then the first three parts of the definition of a metric ([tex]d(p,p) = 0, p\rightarrow d(p,q) > 0, [/tex] and [tex] d(p,q) = d(q,p)[/tex]) are easy enough to prove assuming [tex]xi-yi \in L2[/tex].
I get caught up again in proving that d(p,q) < d(p,r) + d(r,q).
I tried squaring both sides and distributing the summation, but found the resulting right hand side of the equation did not really simplify in a useful way. I'm not really sure where to go from there... any ideas for either of these roadblocks would be greatly appreciated. Thanks.
 
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  • #2
Your missing ingredients are various facets of the 'Cauchy-Schwarz inequality'. Look that up.
 

Related to Infinite sum of squares converges

1. What is an infinite sum of squares?

An infinite sum of squares is a mathematical series that consists of an infinite number of terms, where each term is the square of a number.

2. How is the convergence of an infinite sum of squares determined?

The convergence of an infinite sum of squares is determined by the limit of the sum as the number of terms approaches infinity. If this limit exists and is a finite number, then the sum is said to converge.

3. What does it mean for an infinite sum of squares to converge?

When an infinite sum of squares converges, it means that the sum of all the terms in the series is a finite number. In other words, as more and more terms are added to the sum, the total value approaches a specific number.

4. What are some examples of infinite sums of squares that converge?

One example is the infinite sum of squares of the reciprocals of positive integers, also known as the Basel problem. Another example is the infinite sum of squares of the reciprocals of powers of 2, known as the geometric series.

5. Why is the convergence of infinite sums of squares important?

The convergence of infinite sums of squares is important in many areas of mathematics and physics, including calculus, number theory, and quantum mechanics. It allows for the calculation of otherwise difficult or impossible problems, and helps to understand the behavior of infinite series and their applications.

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