Question

A map \(f: E^{n} \rightarrow E^{1}\) is called a linear functional iff $$ \left(\forall \bar{x}, \bar{y} \in E^{n}\right)\left(\forall a, b \in E^{1}\right) \quad f(a \bar{x}+b \bar{y})=a f(\bar{x})+b f(\bar{y}) $$ Show by induction that \(f\) preserves linear combinations; that is, $$ f\left(\sum_{k=1}^{m} a_{k} \bar{x}_{k}\right)=\sum_{k=1}^{m} a_{k} f\left(\bar{x}_{k}\right) $$ for any \(a_{k} \in E^{1}\) and \(\bar{x}_{k} \in E^{n}\).

Short answer

By induction, a linear functional \( f \) preserves linear combinations for any finite collection of vectors.

Step-by-step solution

Base Case

Consider the case where \( m = 1 \). For a single vector \( \bar{x}_1 \) with scalar \( a_1 \), we want to verify that:\[ f(a_1 \bar{x}_1) = a_1 f(\bar{x}_1) \]This is directly given by the definition of a linear functional. Therefore, the base case holds.

Inductive Hypothesis

Assume that the statement holds true for some arbitrary integer \( m \). That is, suppose:\[ f\left(\sum_{k=1}^{m} a_{k} \bar{x}_{k}\right) = \sum_{k=1}^{m} a_{k} f\left(\bar{x}_{k}\right) \]is true.

Inductive Step

We need to show that, based on the inductive hypothesis, the statement also holds for \( m + 1 \). Consider:\[ f\left( \sum_{k=1}^{m+1} a_{k} \bar{x}_{k} \right) = f\left( \left( \sum_{k=1}^{m} a_{k} \bar{x}_{k} \right) + a_{m+1} \bar{x}_{m+1} \right) \]By the linearity of \( f \):\[ f\left( \sum_{k=1}^{m+1} a_{k} \bar{x}_{k} \right) = f\left( \sum_{k=1}^{m} a_{k} \bar{x}_{k} \right) + f(a_{m+1} \bar{x}_{m+1}) \]Using the inductive hypothesis and linearity:\[ = \sum_{k=1}^{m} a_{k} f(\bar{x}_k) + a_{m+1} f(\bar{x}_{m+1}) \]Which shows:\[ = \sum_{k=1}^{m+1} a_{k} f(\bar{x}_k) \]

Conclusion

Having shown the base case and proved the inductive step, by induction, we conclude that the statement holds for all \( m \geq 1 \). Thus, a linear functional \( f \) indeed preserves linear combinations.

Key concepts

Induction Proof

In mathematics, induction is a powerful method used to prove statements that hold for an infinite number of cases. It comprises two main steps:

• Base Case: Verify the statement for an initial case, usually when the value is smallest, like when \( m = 1 \).
• Inductive Step: Assume the statement is true for a general case \( m \), and then show it also holds for the next case, \( m + 1 \).

The idea is akin to a domino effect—if you can knock over the first domino (base case) and each domino knocks over the next (inductive step), all dominoes will eventually fall, meaning the statement holds for all cases. In the context of the linear functional exercise, we begin by proving the base case for a single vector, \( \bar{x}_1 \), ensuring that the linearity condition is inherently satisfied based on definition. Once the base case is established, we assume it holds for any combination of \( m \) vectors, and then prove it for \( m+1 \) vectors. Successfully completing this, we conclude the statement is true for any number of vectors. Induction is particularly useful in proving properties across infinite sequences or structures.

Linearity

Linearity is a crucial concept in mathematical analysis and functional analysis. A function or map is said to be linear if it satisfies two main properties for all suitable vectors: additivity and homogeneity of degree 1. In simpler terms:

• Additivity: The function applied to the sum of two vectors equals the sum of the function applied to each vector separately. Mathematically, this is: \[ f(\bar{x} + \bar{y}) = f(\bar{x}) + f(\bar{y}) \]
• Homogeneity: The function applied to a scalar multiplication of a vector equals the scalar multiplied by the function of the vector: \[ f(a \bar{x}) = a f(\bar{x}) \]

These properties ensure that linear functions are straightforward and predictable, making them fundamental tools in mapping vector spaces. In the exercise, we utilize these core properties to demonstrate how a linear functional behaves when dealing with a combination of vectors. This showcases the powerful nature of linearity where even complex-looking expressions simplify neatly under these rules.

Mathematical Analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, and measure. It's concerned with the rigorous arguments concerning the stability and convergence of sequences and functions. In the context of linear functionals, mathematical analysis helps elucidate how functions behave under various operations. When we approach an analysis problem, it's essential to rigorously demonstrate properties and behaviors of functions or series. By using methods like proof by induction, we ensure that statements about infinite sequences or structures are logically consistent and universally applicable. Understanding the behavior of linear functionals within mathematical analysis involves dissecting how these functionals operate over vector spaces. We employ algebraic manipulation and analysis techniques to scrutinize their structural properties, ensuring consistency across all operations defined in the space. Analyzing these properties in depth helps solidify the foundational mathematical principles needed for broader applications in fields such as functional analysis and quantum mechanics.