MTH405 Quiz (Hints) - Quiz(Hints MTH-405A(1 Why `p with |x|p:= X |xi |p is a normed linear space for 1 p <[5 points i=1(2 Give an example of a normed

MTH405 Quiz (Hints) - Quiz(Hints MTH-405A(1 Why `p with...

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Quiz (Hints) MTH-405A (1) Why p with || x || p := X i =1 | x i | p ! is a normed linear space for 1 p < ? [5 points] (2) Give an example of a normed linear space X with two norms on it which are not equivalent. Justify your answer. [5 points] Done in class. (3) Show that the range of a bounded linear map may not be always closed. [5 points] Hint: consider the function F : as F ( x n ) = x n n . This is a linear map whose range is not closed because consider the set of sequences y n := (1 , 1 2 , 1 3 , · · · , 1 n , 0 , · · · ). Each of these y n ’s are in the range set because the inverse image of this is the set of sequences z n = (1 , 2 , 3 , · · · , n, 0 , · · · ). Notice that the full sequence ( 1 n ) is the limit of the y n in the sup-norm. So suppose range were a closed subspace, then the inverse image of this full sequence should be in and certainly the inverse image is the full sequence ( n ) / .
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