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May 25th, 2014, 02:18 AM  #1 
Senior Member Joined: Apr 2014 From: zagreb, croatia Posts: 234 Thanks: 33 Math Focus: philosophy/found of math, metamath, logic, set/category/order/number theory, algebra, topology  Direct sum of Banach spaces
Theorem. Let F = ($X_j$, $j$ elem of J) be a family of Banach spaces over field $R$ or $C$ and let 1<=p<$\infty$. With X we note the set of all functions x from the index set J into the union of sets $X_j$ with the following properties: 1. x(j) elem of $X_j$, $j$ elem of J 2. sum j € J of pth powers of norms of vectors x(j) < &\infty$. If we equip X with the usual structure of function vector space, then X is a Banach space relative to the norm (1/p)th power of the sum in 2. X is called direct sum of family ($X_j$, $j$ elem of J) of Banach spaces and is noted $l_p$(F). I wonder what the index set is, since we sum reals or complex numbers. 
May 25th, 2014, 02:26 AM  #2 
Senior Member Joined: Apr 2014 From: zagreb, croatia Posts: 234 Thanks: 33 Math Focus: philosophy/found of math, metamath, logic, set/category/order/number theory, algebra, topology 
2. sum j € J of pth powers of norms of vectors x(j) < infinity. If we equip X with the usual structure of function vector space, then X is a Banach space relative to the norm (1/p)th power of the sum in 2. X is called direct sum of the family. I wonder what the index set is, because we sum real or complex numbers. 
May 25th, 2014, 08:09 AM  #3 
Senior Member Joined: Apr 2014 From: zagreb, croatia Posts: 234 Thanks: 33 Math Focus: philosophy/found of math, metamath, logic, set/category/order/number theory, algebra, topology 
I guess J is N or a finite set. Proof. We first have to show X is a vector space. For x, y elem X, z = x + y and $\lambda$x are defined as the sum and product of function and $\lambda$. Obviously x elem X implies $\lambda$x elem X (because $X_j$ is a vector space) 

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banach, direct, spaces, sum 
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