91Ó°ÊÓ

Functional analysis is a branch of mathematics focused on studying vector spaces and linear functions, particularly in the context of infinite-dimensional spaces. In this exercise, we are working with the finite-dimensional space \(\mathbb{R}^3\) to understand how a linear functional can be extended from a subspace, \(\mathbb{R}^2\), to a larger domain.
Functional analysis often investigates properties of functions that map between vector spaces, such as continuity, linearity, and boundedness. In our exercise, the function \(f(x) = 4\xi_1 - 3\xi_2\) is a linear functional defined on \(\mathbb{R}^2\). The task is to extend this functional linearly to another vector space, \(\mathbb{R}^3\), while preserving its essential linear characteristics.
This concept is deeply rooted in functional analysis, as it requires understanding how functions behave across dimensions. By considering linear extensions, we can gain insights into how vector spaces can be linked through linear mappings.

A subspace is a vector space that is contained within another vector space. In our exercise, we consider a specific subspace of \(\mathbb{R}^3\), where vectors are constrained by the condition \(\xi_3 = 0\).
The subspace \(\mathbb{R}^2\) can be visualized as a plane within the three-dimensional space \(\mathbb{R}^3\). All vectors within this subspace have the form \((\xi_1, \xi_2, 0)\), meaning they do not extend in the direction of the \(\xi_3\) axis.
Subspace properties are crucial as they maintain all vector operations, such as addition and scalar multiplication. This means any linearly extended function must act the same on the subspace as the original function. Thus, the functional \(f(x)\) defined on \(\mathbb{R}^2\) should seamlessly integrate into the larger space \(\mathbb{R}^3\) without loss of its properties.

Linearity is a fundamental property that describes a function or transformation that satisfies two main properties: additivity and homogeneity. In simpler terms, a linear function will preserve vector addition and scalar multiplication.
For our exercise, linearity implies that the function \(f(x) = 4\xi_1 - 3\xi_2\) satisfies:

• Additivity: \(f(x + y) = f(x) + f(y)\) for all vectors \(x\) and \(y\).
• Homogeneity: \(f(\alpha x) = \alpha f(x)\) for any scalar \(\alpha\).

When extending this function to \(\mathbb{R}^3\), with \(\tilde{f}(\xi_1, \xi_2, \xi_3) = 4\xi_1 - 3\xi_2 + a\xi_3\), the linearity of \(f\) is retained because the function continues to respect these properties for all choices of \(a\).
This linearity ensures that even though \(a\) can vary, the extended function \(\tilde{f}\) behaves consistently with the original function's definitions, reinforcing the seamless transition from \(\mathbb{R}^2\) to \(\mathbb{R}^3\).