1. Let X = C¹([0, 1]) and Y = C([0, 1]) be normed vector spaces, both equipped with the uniform norm. Let T: X→Y be a ma

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1 Let X C 0 1 And Y C 0 1 Be Normed Vector Spaces Both Equipped With The Uniform Norm Let T X Y Be A Ma 1 (12.92 KiB) Viewed 39 times

1 Let X C 0 1 And Y C 0 1 Be Normed Vector Spaces Both Equipped With The Uniform Norm Let T X Y Be A Ma 2 (17.07 KiB) Viewed 39 times

these two questions are almost same but little different.
1. Let X = C¹([0, 1]) and Y = C([0, 1]) be normed vector spaces, both equipped with the uniform norm. Let T: X→Y be a map defined by T(f) = f'. Prove that the map T: X→Y is linear but not bounded.
6. Let X = C¹([0, 1]) and Y = C([0, 1]) be normed vector spaces, both equipped with TE- the uniform norm. Let T: XY be a linear map defined by T(f) = f'. Prove that the graph of the map T is closed but T is not bounded. TC)-