For a representation for a bounded linear functionals in L^p, I see we have Riesz's Representation. But how can we find a representation for the bounded linear functionals in l^p for p in [1, infinity)? Someone showed me a representation, but it looked so complicated and big which I didn't understand at all! So, Please I need help!
Also, I don't understand what's the difference between Lp and lp? How the representation will be different from each other in both spaces?
For $x^{\ast}\in(l^{p})^{\ast}$, $p\in[1,\infty)$, let $\beta_{j}=x^{\ast}(e_{j})$, one checks that $x^{\ast}(\alpha)=\displaystyle\sum_{j}\alpha_{j}x^{\ast}e_{j}=\sum_{j}\alpha_{j}\beta_{j}$. It suffices to show that $\beta=(\beta_{j})\in l^{q}$.
Find a $\gamma_{j}$ such that $|\gamma_{j}|=1$ and $|\beta_{j}|=\gamma_{j}\beta_{j}$, then for $1<p<\infty$, \begin{align*} \sum_{j}^{k}|\beta_{j}|^{q}&=\sum_{j=1}^{k}\gamma_{j}\beta_{j}|\beta_{j}|^{q-1}\\ &=x^{\ast}\left(\sum_{j=1}^{k}\gamma_{j}|\beta_{j}|^{q-1}e_{j}\right)\\ &\leq\|x^{\ast}\|\left\|\sum_{j=1}^{k}\gamma_{j}|\beta_{j}|^{q-1}e_{j}\right\|_{p}\\ &=\|x^{\ast}\|\left(\sum_{j=1}^{k}|\beta_{j}|^{q}\right)^{1/p}, \end{align*} now taking $k\rightarrow\infty$ to conclude.
For $p=1$, we have $|\beta_{j}|=|x^{\ast}(e_{j})|\leq\|x^{\ast}\|\|e_{j}\|_{1}=\|x^{\ast}\|$.