91Ó°ÊÓ

The operator norm is a crucial concept in functional analysis, especially when dealing with linear maps between normed vector spaces. It is defined as the supremum (or the least upper bound) of the absolute value of the function over all input vectors of a given norm.
This norm helps to gauge the 'size' of a linear map in a way that is consistent with the norms of its inputs and outputs. In mathematical terms, the operator norm \( \|f\| \) of a function \( f(x) \) is given by the formula:
\[ \|f\| = \sup_{\|x\|_p \leq 1} |f(x)| \]
This formula indicates that we evaluate the maximum effect of the function \( f \) on any vector \( x \) whose \( p \)-norm is less than or equal to 1.

Hölder's inequality is a fundamental result in measure theory and is widely used in various branches of mathematics, including functional analysis. It provides a bound for the integral (or sum) of the product of two functions and is particularly useful for dealing with norms.
Formally, for any sequences of real or complex numbers \( (a_i) \) and \( (b_i) \), the inequality can be expressed as:
\[ \left| \sum_{i=1}^n a_i b_i \right| \leq \left( \sum_{i=1}^n |a_i|^p \right)^{1/p} \left( \sum_{i=1}^n |b_i|^q \right)^{1/q} \]
where \( 1 \leq p, q \leq \infty \) and \( 1/p + 1/q = 1 \).
In the context of the given problem, Hölder's inequality is applied to find an upper bound for \( |f(x)| \) by setting \( a_i = u(i) \) and \( b_i = x(i) \).

The concept of conjugate exponents comes into play when dealing with Hölder's inequality. The conjugate exponent \( q \) of a given exponent \( p \) satisfies the relation:
\[ \frac{1}{p} + \frac{1}{q} = 1 \]
For example, if \( p = 2 \), then the conjugate exponent \( q \) is 2 because \( 1/2 + 1/2 = 1 \). Similarly, if \( p = 1 \), then \( q \) is \( \infty \), and vice versa.
This relationship helps in determining the appropriate norms and applying Hölder's inequality correctly. In the exercise, we need to use \( q \)-norm, which is the conjugate exponent of the \( p \)-norm that the space \( \mathbb{K}^n \) is endowed with.

The \( p \)-norm is a generalization of various norms used to measure the length or size of vectors in \( \mathbb{K}^n \). The \( p \)-norm \( \|x\|_p \) of a vector \( x \) is defined as:
\[ \|x\|_p = \left( \sum_{i=1}^n |x(i)|^p \right)^{1/p} \]
for \( 1 \leq p \leq \infty \).
For \( p = 1 \), this becomes the Manhattan (or taxicab) norm.
For \( p = 2 \), it turns into the Euclidean norm.
For \( p = \infty \), it becomes the maximum norm defined as \( \|x\|_\infty = \max_i |x(i)| \).
Understanding these norms is essential for computing operator norms and applying Hölder's inequality as shown in the solution steps.

A linear map (or linear transformation) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. Given a vector space \( \mathbb{K}^n \), a linear map \( f: \mathbb{K}^n \to \mathbb{K} \) can be represented as:
\[ f(x) = u(1)x(1) + u(2)x(2) + \cdots + u(n)x(n) \]
where \( u \in \mathbb{K}^n \) and \( x \in \mathbb{K}^n \).
Linear maps are fundamental to many areas of mathematics as they simplify the structure and analysis of vector spaces. In this exercise, the linear map \( f \) defined by \( u \) helps in establishing the relationship between the operator norm \( \|f\| \) and the \( q \)-norm \( \|u\|_q \).