Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm trying to show that $\ell_{p}$ is a vector space for any $1\leqslant p<\infty$ . So given two infinite series $\left(x_{n}\right)_{n=1}^{\infty}$ and $\left(y_{n}\right)_{n=1}^{\infty}$ such such that ${\displaystyle \sum_{n=1}^{\infty}\left|x_{n}\right|^{p}}$ and ${\displaystyle \sum_{n=1}^{\infty}\left|y_{n}\right|^{p}}$ converge and $\alpha,\beta\in\mathbb{R}$ I want to show that $$\left[\alpha\left(x_{n}\right)_{n=1}^{\infty}+\beta\left(y_{n}\right)_{n=1}^{\infty}\right]\in\ell_{p}$$ To do that I need to show that $${\displaystyle \sum_{n=1}^{\infty}\left|\alpha x_{n}+\beta y_{n}\right|^{p}}<\infty$$ For some reason I'm finding this quite hard, I've been unsuccesful in bounding this sum from above by the two convergent series I started with.

share|improve this question
    
You should add the homework tag. –  tst Jan 5 '13 at 14:18

2 Answers 2

up vote 0 down vote accepted

You are actually trying to prove it's a subspace of a bigger vector space, for which we know it's already one. For example, the space of real sequences. (otherwise, we have to check all the axioms).

Hint: use and show the inequality $(a+b)^p\leqslant 2^{p-1}(a^p+b^p)$ for non-negative $a$ and $b$.

share|improve this answer
1  
At first I wasn't sure how to justify that inequality but after the pointer from Learner I recalled that $x^p$ is convex over the positive reals and that sorted it all out. Thanks for both your help. –  Serpahimz Jan 5 '13 at 14:45

You could look at the proof of the Minkowski inequality (the sequence case).

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.