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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 p-th 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$(F). I wonder what the index set is, since we sum reals or complex numbers.  May 25th, 2014, 02:26 AM #2 Senior Member Joined: Apr 2014 From: zagreb, croatia Posts: 234 Thanks: 33 Math Focus: philosophy/found of math, metamath, logic, set/category/order/number theory, algebra, topology 2. sum j € J of p-th powers of norms of vectors x(j) < infinity. If we equip X with the usual structure of function vector space, then X is a Banach space relative to the norm (1/p)-th power of the sum in 2. X is called direct sum of the family. I wonder what the index set is, because we sum real or complex numbers.  May 25th, 2014, 08:09 AM #3 Senior Member Joined: Apr 2014 From: zagreb, croatia Posts: 234 Thanks: 33 Math Focus: philosophy/found of math, metamath, logic, set/category/order/number theory, algebra, topology I guess J is N or a finite set. Proof. We first have to show X is a vector space. For x, y elem X, z = x + y and$\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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