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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) Tags banach, direct, spaces, sum Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Cogline Real Analysis 3 January 28th, 2010 01:53 PM otaniyul Real Analysis 2 August 24th, 2009 09:01 PM aptx4869 Real Analysis 3 October 25th, 2008 02:13 PM Nevski Real Analysis 1 February 26th, 2008 10:33 AM mathiask Linear Algebra 4 October 15th, 2007 06:26 AM

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