# Prove that l^p is a subset of l^q for all p,q from 1 to infinity

• cbarker1
In summary, the conversation discusses a problem with applying an inequality for a finite series and proving the relationship between p-norm and q-norm. There may be a condition on p and q, and it is impossible to prove that l^p is strictly less than l^q and l^q is strictly less than l^p. The conversation also mentions the relationship between the sums of two sequences of non-negative numbers with specific properties.

#### cbarker1

Gold Member
MHB
Homework Statement
Prove that l^p is a subset of l^q for all p,q from 1 to infinity. Then prove it is strict subset. First, prove that a^t<=a for all t,a in (0,1]. Then prove that finite sum of |x_i|^t<= the sum of |xi|.
Relevant Equations
a^t<=a for all a,t
p-norm's definition.
Dear everyone,

I am having trouble with this problem. I have convinced myself that the ##a^t-a\leq 0## is true. Now, I am trying to applying this inequality for the finite series and I don't know where to start. After that, proving that the p-norm is less or equal to the q-norm.

Thanks,
Cbarker1

Is there a condition on $p$ and $q$, such as $q < p$? Otherwise you are being asked to prove $l^p \subsetneq l^q \subsetneq l^p$ which is impossible.

If you have two sequences of non-negative numbers, with the property that each element of the first sequence is less than or equal to the corresponding element of the second sequence, what can you say about the sums of those sequences?

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## 1. What is l^p and l^q?

l^p and l^q are mathematical notations for function spaces, specifically for sequences of numbers. The notation l^p represents the space of all sequences whose p-th power is summable, while l^q represents the space of all sequences whose q-th power is summable.

## 2. What does it mean for l^p to be a subset of l^q?

For l^p to be a subset of l^q means that every sequence in l^p is also in l^q. In other words, every sequence whose p-th power is summable will also have a q-th power that is summable.

## 3. Why is it important to prove that l^p is a subset of l^q?

This proof is important because it shows the relationship between different function spaces and their properties. It also helps in understanding the convergence and divergence of sequences in these function spaces.

## 4. How can we prove that l^p is a subset of l^q?

We can prove this by using the Hölder's inequality, which states that for any two positive numbers, p and q, with 1/p + 1/q = 1, the product of two sequences in l^p and l^q is also in l^1. By using this inequality, we can show that if a sequence is in l^p, then it must also be in l^q.

## 5. Does this proof hold for all values of p and q from 1 to infinity?

Yes, this proof holds for all values of p and q from 1 to infinity. This is because the Hölder's inequality is applicable for all values of p and q, as long as they are positive numbers and 1/p + 1/q = 1.