I know that a bounded linear functional has these two properties:
for $A:V \rightarrow R$ and $V$ being a vectorspace we have:
(1) $A(x+y) = A(x) + A(y)$, for all $x,y \in V$
(2) $A(\alpha x) = \alpha A(x)$, for all $x \in V$ and $\alpha \in R$
(3) $|A(x)| \le C.||x||$, for all $x \in V$ and $C > 0$
To my Problem:
i need to show that $A_b:l^p(N)→R,(a_n)_{n∈N}\rightarrow \sum_{n=1}^\inf a_n b_n$
,with $b = (b_n)_{n\in N} $ and $p,q > 1$ with $\frac{1}{p}+\frac{1}{q}=1$
is defined as a bounded linear functional on $(l^p(N),||.||_p)$.
I am trying to use "Holder inequality" but i was not successful.
I would appreciate it, if you know how to approache this problem and every other problems like this one.
Thank you
Linearity is obvious. Boundedness follows as you said by Holder's inequality:
$$|A_b(a_n)|=\bigg|\sum_{n=1}^\infty a_nb_n\bigg|\leq\sum_{n=1}^\infty|a_n||b_n|\leq\bigg(\sum_{n=1}^\infty|a_n|^p\bigg)^{1/p}\cdot\bigg(\sum_{n=1}^\infty|b_n|^q\bigg)^{1/q}=\|(a_n)\|_p\cdot \|(b_n)\|_q$$
This is true for all $(a_n)\in\ell^p$, so, assuming that $(b_n)\in\ell^q$, $A_b$ is a bounded functional and $\|A_b\|\leq\|(b_n)\|_q$. Actually, equality holds, $\|A_b\|=\|(b_n)\|_q$, but what we did above is enough.
edit upon request: Linearity is very simple: let $\lambda,\mu$ be scalars and let $(a_n), (a'_n)$ be sequences in $\ell^p$. Also, let $N\geq1$ be a natural number. Then $$\sum_{n=1}^N(\lambda a_n+\mu a'_n)b_n=\lambda\sum_{n=1}^Na_nb_n+\mu\sum_{n=1}^Na'_nb_n$$ As this is true for any $N$, let $N\to\infty$ to obtain $$\sum_{n=1}^\infty(\lambda a_n+\mu a'_n)b_n=\lambda\sum_{n=1}^\infty a_nb_n+\mu\sum_{n=1}^\infty a'_nb_n$$ i.e. $A_b(\lambda(a_n)+\mu(a'_n))=\lambda A_b(a_n)+\mu A_b(a'n)$, which shows linearity.