Question

show that the absolute value of a linear functional has the subadditive and positive homogeneous properties.

Answer 1

Answer on Question #70193 – Math – Functional Analysis

Question

Show that the absolute value of a linear functional has the subadditive and positive homogeneous properties.

Solution

Let $f: X \to \mathbb{P}$ be a linear functional and $X$ a vector space. We show that $|f|$ has the subadditive and positive homogeneous properties.

Consider arbitrary $x, y \in X$. Since $f$ is a linear functional, $f$ is an additive function. Thus, $f(x + y) = f(x) + f(y)$. Then

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

So, $|f(x + y)| \leq |f(x)| + |f(x)|$ for each $x, y \in X$. This means that $f$ has the subadditive property.

Consider any $x \in X$ and $\alpha \in \mathbb{P}$, $\alpha > 0$. Since $f$ is a linear functional, $f$ is homogeneous. Thus, $f(\alpha x) = \alpha f(x)$. Then

$$|f(\alpha x)| = |\alpha f(x)| = |\alpha| |f(x)| = \alpha |f(x)|$$

because $\alpha > 0$. So, $|f(\alpha x)| = \alpha |f(x)|$ for each $x \in X$ and $\alpha \in \mathbb{P}$, $\alpha > 0$. This means that $f$ has the positive homogeneous property.

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