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Sabita Vabi Bangla Pdf 45
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Showing that the superposition of a countably infinite family of $L_1$ spaces is an $L_1$ space?
Let $ \left\{ X_i \right\}_{i\in I}$ be a collection of Banach Spaces such that $X_i\subset X_j$ for all $i\leq j$. Show that the direct sum $\bigoplus_{i\in I} X_i$ becomes a Banach space in the $L_1$-metric when you consider the norm $||\sum_{i\in I}f_i||=\sum_{i\in I}||f_i||$.
The direct sum is the set of all finite linear combinations $\sum_{i\in I}f_i$. As usual there’s a funny book with that name, but the question is not exactly from that one, since that one is not a mathematical one. I’ve never studied linear algebra/functional analysis or measure theory yet.
Edit:
I know how to do it using the countable additivity property, but now I wonder if it is possible to generalize that proof to all countably infinite spaces.
A:
Each finite linear combination $x=\sum_{i=1}^n f_i$ has norm
$$||x||^2=\sum_{i=1}^n ||f_i||^2.$$
Now, note that you can write
$$\sum_{i\in I}||f_i||=\sum_{i=1}^\infty||f_i||=||\sum_{i=1}^\infty f_i||\leq\sum_{i=1}^\infty||f_i||^2
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