Problem about problem solving I am having some problems on how to solve problems. When I read a chapter on say group theory or real analysis, I feel that I have understood the concepts quite well, but when I start solving exercises, after 5 or 6 problems I get stuck. When I see a tough problem, I get messed and cannot find how to come out of it. When I see the solution I have no problem in understanding it. I think about many things but cannot come up with a solution. Is it a problem of my basics not being right? I have an entrance exam &amp; being in this situation is frustrating for me. Please help.
Well, it's possible that the books your reading have really bad exercises. But if this happens often, the more likely problem is that your feeling that you have grabbed the concepts quite well must be mistaken. 

As useful test, when you look at a problem, is to ask yourself "Can I define precisely every single term used in this problem? " And then "What theorems do I know about these things?" If you can't recite the theorems -- with all the hypotheses! -- then you're probably not going to be able to do the problem. It's like trying to build a cabinet while you keep forgetting there's a saw in your toolbox. 

I know that's not a lot of help, but it's the best I can do without more detail.
suppose {T} is a collection of bounded operators on a Hilbert space H, with  IT&lt;1   for all k. Suppose also that.
Let SN - Σ Ν T k =-Ν
Show that SNf)     converges as N00 , for every Є Н . If  Tf denotes the limit, prove that  |T||&lt; 1 . [Hint: Consider first the case when only finitely many of the  TK are non-zero, and note that the ranges of the TK are mutually orthogonal.]