Answer :
Answer:
The Hahn-Banach Theorem is a fundamental result in functional analysis. The correct statement regarding the theorem is:
A) f can be extended to the whole space without increasing its norm.
Step-by-step explanation:
To explain further, the Hahn-Banach Theorem states that if you have a sublinear functional p defined on a vector space V and a linear functional f defined on a subspace[tex]\( U \subseteq V \) such that \( f(u) \leq p(u) \) for all \( u \in U \), then \( f \) can be extended to a linear functional \( \tilde{f} \) on the entire space \( V \) in such a way that \( \tilde{f}(v) \leq p(v) \) for all \( v \in V \).[/tex]
In the context of normed spaces, if f is a bounded linear functional defined on a subspace U of a normed space V, the Hahn-Banach Theorem guarantees that f can be extended to a bounded linear functional on V with the same norm. This means the extension does not increase the norm of f.
The Hahn-Banach Theorem and its implications on a normed space regarding a sublinear functional and a linear functional. To address the options given:
- (A) (f) can be extended to the whole space without increasing its norm: It is true. The Hahn-Banach Theorem guarantees that if you have a linear functional defined on a subspace of a normed space, it can be extended to the entire space without increasing its norm.
- (B) (f) is uniformly continuous: It is not necessarily guaranteed by the Hahn-Banach Theorem.
- (C) (p) is continuous: It is not directly guaranteed by the theorem.
- (D) The dual space is: It is incomplete and not directly related to the Hahn-Banach Theorem.
- The Hahn-Banach Theorem allows extending a linear functional to the whole space without increasing its norm, maintaining its integrity. It is a fundamental concept in functional analysis and space expansion through functionals.
Hahn-Banach Theorem:
- The Hahn-Banach Theorem states that given a sublinear functional (p) and a linear functional (f) defined on a subspace of a normed space, the linear functional (f) can be extended to the whole space without increasing its norm. This theorem plays a crucial role in functional analysis and is used to extend functionals defined on subspaces to the whole space while preserving certain properties.
- The Hahn-Banach Theorem enables the extension of a linear functional to the whole space without increasing its norm. Options B, C, and D are not directly ensured by the theorem. It is a crucial concept in functional analysis within normed spaces.
Understanding the Hahn-Banach Theorem
- The Hahn-Banach Theorem is a fundamental result in functional analysis, which allows the extension of bounded linear functionals. This theorem is essential in studying normed spaces and has significant implications, particularly in expanding the scope of analysis within these spaces.
- For example, if you have a subspace V of a normed space X and a linear functional f: V → R that is dominated by a sublinear functional p on X, then there exists an extension F: X → R of f such that F ≤ p.
