Q1. State whether the following statements True or False? Justify your answers:
- (a)) The function || || defined on R" as: ||x|| = ̑̑̑̑̑̑j=1 ̑j for x = (a,a2,....,a)∈ R" is a norm. (100 words)
- (b)) Co is a Banach space. (100 words)
- (c)) If A is the right shift operator on l, then the eigen spectrum is non-empty. (100 words)
- (d)) If a normed linear space is reflexive, then so is its dual space. (100 words)
- (e)) If a normed linear space X is finite dimensional, then so is X'. (100 words)
Key Points:
- A function must satisfy non-negativity, definiteness, homogeneity, and triangle inequality to be a norm.
- The space c₀ (sequences converging to zero) is a Banach space under the supremum norm.
- The right shift operator on l² has an empty eigen spectrum (no eigenvalues).
- A normed linear space is reflexive if and only if its dual space is reflexive.
Answer: This question assesses fundamental concepts in Functional Analysis, including properties of norms, completeness of spaces, spectral theory of operators, and properties of dual and reflexive spaces. Each sub-question requires an understanding of definitions and key theorems to determine the truthfulness of the statement and provide a concise justification. These concepts are integral to understanding the structure and behavior of infinite-dimensional vector spaces, which are central to the stud...