Question

show that absolute value of. a linear functional has properties of sublinear functional

Answer 1

Answer on Question #81212 – Math – Functional Analysis

Question

show that absolute value of. a linear functional has properties of sublinear functional.

Solution

Let $f$ be a linear functional $f: X \to \mathbb{R}$.

Consider $\varphi(x) = \|f(x)\|$.

We have for $\lambda \geq 0$ :

$$\varphi(\lambda x) = \|f(\lambda x)\| = \|\lambda f(x)\| = \lambda \|f(x)\| = \lambda \varphi(x).$$

This proves that $\varphi$ is nonnegatively homogeneous.

Then, for $x, y \in X$ :

$$\varphi(x + y) = \|f(x + y)\| = \|f(x) + f(y)\| \leq \|f(x)\| + \|f(y)\| = \varphi(x) + \varphi(y).$$

This proves that $\varphi$ is subadditive.

These two properties prove that $\varphi$ is sublinear functional.

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