WWGD
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I understood it a bit differently. The norm of the operator is the sup over the unit ball. We normalize a countable subset of the basis so that ||e_i||=1 . Then we declare T(e_i)=i (Standard norm on the Reals/Image), so that the Sup of the images goes to ## \infty ##. But I like your way too, to have more than one approach to see it be more understandable.mathwonk said:me too, maybe if e1,e2,... is an infinite sequence of unit length independent vectors, and T(ej) = j, then although ej/j-->0, T(ej/j) = 1, for all j.
(It seems I still know Banach spaces better than I know trig and one vbl calc.)
My conclusion was that since, given how I was taught, I did not understand anything, if one wants to understand, one should be taught the opposite to how I was. Of course that also may not be sufficient. Maybe it was actually my fault that I did not think and work hard enough, ask questions, look for examples? Nahhh...