Modulus of Linear Functional on Vector Space is Seminorm
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Theorem
Let $\GF \in \set {\R, \C}$.
Let $X$ be a vector space over $\GF$.
Let $f : X \to \GF$ be a linear functional.
Define $p_f : X \to \R_{\ge 0}$ by:
- $\map {p_f} x = \cmod {\map f x}$
for each $x \in X$.
Then $p_f$ is a seminorm.
Proof
Proof of Seminorm Axiom $\text N 2$: Positive Homogeneity
For each $\lambda \in \GF$ and $x \in X$, we have:
\(\ds \map {p_f} {\lambda x}\) | \(=\) | \(\ds \cmod {\map f {\lambda x} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {\lambda \map f x}\) | since $f$ is linear | |||||||||||
\(\ds \) | \(=\) | \(\ds \cmod \lambda \cmod {\map f x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cmod \lambda \map {p_f} x\) |
$\Box$
Proof of Seminorm Axiom $\text N 3$: Triangle Inequality
For each $x, y \in X$, we have:
\(\ds \map {p_f} {x + y}\) | \(=\) | \(\ds \cmod {\map f {x + y} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {\map f x + \map f y}\) | since $f$ is linear | |||||||||||
\(\ds \) | \(\le\) | \(\ds \cmod {\map f x} + \cmod {\map f y}\) | Triangle Inequality for Real Numbers if $\GF = \R$ and Triangle Inequality for Complex Numbers if $\GF = \C$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {p_f} x + \map {p_f} y\) |
$\blacksquare$