MT-301 : Functional Analysis - Mathematics Assignment Help

Assignment Task :

Task:

Attempt any five of the following:
Q.1 Define normed linear spaces .If { X ,|| .|| X } and { Y,|| .||y } are normed space prove that || (x,y) ||= max{ ||x||x , ||Y||Y } ,(X,Y)  X × Y .
Q.2 If the real linear space X= C1 =[0,1] Of all continuously differentiable function define on [0,1] with the norm, ||f||=sup|f(x)|, prove that ( X , ||.|| ) is incomplenormed space. X  [0,1]

Q.3 If N be a non zero normed linear space proved that N is a Branch space if and only if { X  N : ||X|| = 1 } is complete.
Q.4 If N and N’ are normed linear spaces and T: N → N’ a linear transformation then show that T is continuous if and only if T is bounded.
Q.5 If T is continuous linear transformation of a normed space N into a normed space N’ and if M its null sapce then show that T induced a natural linear transformation T of N/ M into N and ||T ‘ || = || T ||
Q.6 State and prove Hahn- Banach Theorem.

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