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 October 13th, 2010, 09:27 AM #1 Newbie   Joined: Dec 2008 From: Copenhagen, Denmark Posts: 29 Thanks: 0 A problem from functional analysis Hi I'm currently following a first course in functional analysis, and a problem from my weekly assignment is bothering me quite a bit. It would be greatly appreciated if anyone could give me a hint or maybe even go through the steps in the solution. The problem is given below. For a normed vector space , we denote by the space of bounded linear operators . (The problem) Let be a complex vector space with an inner product and assume that . 1) Show that for all if and only if is the zero operator. 2) Show next that for all if and only if is the zero operator. 3) Are these results true if the vector space is a real vector space? October 13th, 2010, 04:00 PM #2 Global Moderator   Joined: May 2007 Posts: 6,822 Thanks: 723 Re: A problem from functional analysis 1. and 2. The if parts are trivially obvious. For 1. If Tx = y ? 0 for some x, the (Tx,y) ? 0. Therefore only if. For 2. Simple example: 2 dimensional space with base vectors a and b, with (a,b)=0. Let Ta=b and Tb=-a, then (Tx,x)=0 for all x, therefore only if is false. I don't see any distinction between real and complex spaces. However you may have to use complex conjugates for complex spaces in the above analysis. October 15th, 2010, 01:42 PM #3 Global Moderator   Joined: May 2007 Posts: 6,822 Thanks: 723 Re: A problem from functional analysis Correction: The analysis for 1. holds for both both real and complex vector spaces. The analysis for 2. is only for real vector spaces. I believe that 2. is true for complex vector spaces, but I haven't been able to work it out completely. October 16th, 2010, 06:00 AM #4 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 Re: A problem from functional analysis Suppose V is complex, and for all Then for all and Expand these out, and you should be able to deduce that for all and therefore (2) reduces to (1). October 20th, 2010, 05:53 AM #5 Newbie   Joined: Dec 2008 From: Copenhagen, Denmark Posts: 29 Thanks: 0 Re: A problem from functional analysis Thanks for responding both of you. I solved the problem in the following way (hopefully no typo's) 1) The if part is trivial. The only if part I show by contraposition. Let be any operater in but not the zero operator. Then there exists such that and so we have . Choose and then we have found such that . Thus must be the zero operator. 2) Again the if part is trivial. The only if part I show by choosing again as any operator in but not the zero operator. Then we know there exists such that . Assume now that . Then we have Thus must be the zero operator. 3) These results are not true if is a real vector space. This is showed by an example. Choose and let be given by for . Then is linear, bounded and not the zero operator. Equip with the standard inner product and then we have for all . Tags analysis, functional, problem 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 AfroMike Real Analysis 6 August 28th, 2013 12:23 PM AfroMike Real Analysis 1 August 21st, 2013 10:57 AM PeterPan Real Analysis 1 April 29th, 2013 11:46 AM Azari123 Academic Guidance 1 August 11th, 2012 12:29 AM kien Real Analysis 4 October 19th, 2008 08:43 AM

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