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

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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.
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cbarker1
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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
 
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Is there a condition on [itex]p[/itex] and [itex]q[/itex], such as [itex]q < p[/itex]? Otherwise you are being asked to prove [itex]l^p \subsetneq l^q \subsetneq l^p[/itex] 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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FAQ: Prove that l^p is a subset of l^q for all p,q from 1 to infinity

What is the definition of \( l^p \) and \( l^q \) spaces?

The \( l^p \) space is defined as the set of all sequences \( (x_n) \) of real or complex numbers such that the series \( \sum_{n=1}^\infty |x_n|^p \) converges, where \( p \geq 1 \). Similarly, \( l^q \) space is defined for sequences where \( \sum_{n=1}^\infty |x_n|^q \) converges, where \( q \geq 1 \). These spaces are equipped with the norm \( \|x\|_p = \left( \sum_{n=1}^\infty |x_n|^p \right)^{1/p} \) for \( l^p \) and \( \|x\|_q = \left( \sum_{n=1}^\infty |x_n|^q \right)^{1/q} \) for \( l^q \).

Why is \( l^p \) not always a subset of \( l^q \)?

Whether \( l^p \) is a subset of \( l^q \) depends on the relationship between \( p \) and \( q \). For \( p \leq q \), every sequence in \( l^p \) is also in \( l^q \) because the \( p \)-norm is generally larger or equal to the \( q \)-norm. However, for \( p > q \), there exist sequences that are in \( l^p \) but not in \( l^q \), as the \( q \)-norm might diverge even if the \( p \)-norm converges.

Can you provide an example where \( l^p \) is a subset of \( l^q \)?

Consider \( p = 1 \) and \( q = 2 \). Any sequence \( (x_n) \) in \( l^1 \) satisfies \( \sum_{n=1}^\infty |x_n| < \infty \). Since \( |x_n|^2 \leq |x_n| \) for all \( x_n \leq 1 \), it follows that \( \sum_{n=1}^\infty |x_n|^2 \leq \sum_{n=1}^\infty |x_n| \), which implies that \( (x_n) \in l^2 \). Hence, \( l^1 \subset l^2 \).

Is there a general condition that determines when \( l^p \) is a subset of \( l

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