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Let w(x)>0 for a ≤x ≤b be a weight function. (a) Prove that...

Let $w(x)>0$ for $a \leq x \leq b$ be a weight function. (a) Prove that $\|f\|_{1, w}=\int_a^b|f(x)| w(x) d x$ defines a norm on $\mathrm{C}^0[a, b]$, called the weighted $\mathrm{L}^1$ norm. (b) Do the same for the weighted $\mathrm{L}^{\infty}$ norm $\|f\|_{\infty, w}=\max \{|f(x)| w(x): a \leq x \leq b\}$.

Let $w(x)>0$ for $a \leq x \leq b$ be a weight function. (a) Prove that
$\|f\|_{1, w}=\int_a^b|f(x)| w(x) d x$ defines a norm on $\mathrm{C}^0[a, b]$, called the weighted $\mathrm{L}^1$ norm.
(b) Do the same for the weighted $\mathrm{L}^{\infty}$ norm $\|f\|_{\infty, w}=\max \{|f(x)| w(x): a \leq x \leq b\}$.

Key Concepts

L? Norm

The L? norm, often called the supremum or maximum norm, measures the size of a function by considering the maximum absolute value its outputs can obtain. In the weighted L? norm, each function value is multiplied by a weight and then the maximum is taken. Like other norms, it must be shown to satisfy the required norm properties.

L¹ Norm

The L¹ norm on a function space is defined as the integral of the absolute value of the function over its domain. In the weighted version, the integrand is multiplied by a positive weight function, which can affect the size measurement of functions. Its proof as a norm involves verifying positivity, definiteness, homogeneity, and the triangle inequality through integral properties.

Weight Function

A weight function is a positive function defined on a given domain that is used to modify conventional norms, thereby introducing a variable density or importance across different parts of the domain. Its positivity ensures that the norm properties, especially the zero vector only having zero norm, are maintained in the weighted setting.

Norm

A norm is a function that assigns a non-negative length or size to elements in a vector space, satisfying the properties of positive definiteness, homogeneity (scaling), and the triangle inequality. These properties are essential in defining metrics, convergence, and continuity within the space.

Weighted Norm

A weighted norm is a modification of a standard norm where a weight function is introduced to emphasize or de-emphasize certain parts of the domain. This approach is particularly useful when the importance or behavior of the function varies over the domain, and it still must satisfy the fundamental norm properties.

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