I have a small confusion,i know how to prove $l^2(\mathbb R)$ is a vector space but i am not getting any idea to prove $l^3(\mathbb R)$ is a vector space.
Vector space is defined as
I have a small confusion,i know how to prove $l^2(\mathbb R)$ is a vector space but i am not getting any idea to prove $l^3(\mathbb R)$ is a vector space.
Vector space is defined as
Non-negativity and other norm properties are trivial to prove. The one which may be difficult is the triangle inequality.
To prove the triangle inequality for $\ell^p$ for $1<p<\infty$ (the cases for $p=1,\infty$ are trivial), you should know the following:
Now, you can prove the triangle inquality in $\ell^p$ for $1<p<\infty$ (try $p=1,p=\infty$ on your own): $\sum_i |x_i+y_i|^p = \sum_i |x_i+y_i|^{p-1} |x_i+y_i| \leq \sum_i (|x_i+y_i|^{p-1} |x_i| + |x_i+y_i|^{p-1} |y_i|)$.
Now, apply Holder's inequality to both terms to get the upper bound $(\sum_i |x_i+y_i|^{(p-1)q})^{1/q} (\sum_i |x_i|^p)^{1/p} + (\sum_i |x_i+y_i|^{(p-1)q})^{1/q} (\sum_i |y_i|^p)^{1/p}$. Now, divide through by $(\sum_i |x_i+y_i|^p)^{1/q}$ and note that $(p-1)q=p$ and $1-\frac{1}{q}=\frac{1}{p}$ to get the result that $(\sum_i |x_i|^p)^{1/p}$ is indeed a norm.
Suppose that $\sum_n|x_n|^3<\infty$, $\sum_n|y_n|^3<\infty$. The key inequality is $$ |a+b|^3\leq(|a|+|b|)^3\leq(2\max\{|a|,|b|\})^3=8\max\{|a|^3,|b|^3\}\leq 8(|a|^3+|b|^3). $$
Then $$ \sum_n|x_n+y_n|^3\leq 8\,\sum_n|x_n|^3+8\,\sum_n|y_n|^3<\infty. $$