Teaching JKPSC Lecturership Mock Test Series 2024-25 Mathematical Science Analysis Normed Linear Spaces
Let \(\rm \displaystyle \ell^p=\left\{x=\left(x_n\right)_{n \geq 1}: x_n \in \mathbb{R},\|x\|_p=\left(\sum_{n=1}^{\infty}\left|x_n\right|^p\right)^{1 / p}<\infty\right\}\) for p = 1, 2. Let C00 = {(xn)n≥1 : xn = 0 for all but finitely many n ≥ 1} .
For x = (xn)n≥1 ∈ C00, define \(\rm \displaystyle f(x)=\sum_{n=1}^{\infty} \frac{x_n}{\sqrt{n}}\). Consider the following statements.
I. There exists a continuous linear functional F on (ℓ1, ∥·∥1) such that F = f on C00.
II. There exists a continuous linear functional G on (ℓ2, ∥·∥2) such that G = f on C00.
Which one of the following is correct?
1
Both I and II are TRUE
2
I is TRUE and II is FALSE
3
I is FALSE and II is TRUE
4
Both I and II are FALSE