I have some troubles in proof that $\lambda$-sum of $L_1$ is a banach space, is mensioned in a Wnuk paper that is a banach space but i dont understand how can i proof that is complete.
Let $\lambda \subset \mathbb{R}^{\mathbb{N}}$, such that $\lambda$ is a Banach lattice with the order cordenate-cordente and an ideal of $\mathbb{R}^{\mathbb{N}}$ then define the set: \begin{eqnarray*} \lambda(L^1):=\{(f_n)_n\subset L^1:(\|f_n\|_{L^1})_n\in \lambda\} \end{eqnarray*} proof that $\lambda(L_1)$ is a Banach space with the norm \begin{eqnarray*} \|F\|=\|(\|f_n\|_{L^1})_n\|_\lambda \text{ with }F=(f_n)_n \end{eqnarray*} Defintion: Let $A$ a Riesz space, $B \subset A$ is an ideal iff for all $y\in A$ if there is an $x\in B$ such that $\left| y\right|\leq \left| x\right|$ then $y\in B $.
My atempt:
The only vector space property that is not easy to proof is the sum clausure. Let $F,G \in \lambda(L^1)$ then $F+G=(f_i+g_i)_i$, each $f_i,g_i \in L^1$ then $f_i+g_i\in L^1$ for all $i\in \mathbb{N}$. Note that for all $i$ \begin{eqnarray*} \|f_i+g_i\|_{L^1}\leq \|f_i\|_{L^1}+\|g_i\|_{L^1} \end{eqnarray*} and \begin{eqnarray*} \left| (\|f_i\|_{L^1})_i\right|=(\|f_i\|_{L^1})_i \hspace{0,5cm} \left| (\|g_i\|_{L^1})_i\right|=(\|g_i\|_{L^1})_i \end{eqnarray*} then \begin{eqnarray*} \left| (\|f_i+g_i\|_{L^1})_i\right| \leq \left| (\|f_i\|_{L^1})_i+(\|g_i\|_{L^1})_i\right| \end{eqnarray*} but $\lambda$ is an ideal then $(\|f_i+g_i\|_{L^1})_i \in \lambda$. Then $\lambda(L^1)$ is a vector space. The proof of that $\|F\|$ is a norm is easy, but a proof of him is Banach is to hard we don´t have any information of a norm in $\lambda$ then i can´t start the proof, Can you help me with any hint or any book that mention this space. Thank you.