Teaching JKPSC Lecturership Mock Test Series 2024-25 Mathematical Science Topology Topological spaces and Continuous Functions
Let C[0, 1] be the Banach space of real valued continuous functions on [0, 1] equipped with the supremum norm. Define T : C[0, 1] → C[0, 1] by
\(\displaystyle (T f)(x)=\int_0^x x f(t) d t\).
Let R(T) denote the range space of T. Consider the following statements:
P: T is a bounded linear operator.
Q: T-1 : R(T) → C[0, 1] exists and is bounded.
Then
1
both P and Q are TRUE
2
P is TRUE and Q is FALSE
3
P is FALSE and Q is TRUE
4
both P and Q are FALSE