Inner Product with Vector is Bounded Linear Functional

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Let $\GF$ be a subfield of $\C$.

Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space over $\GF$.

Let $v_0 \in V$.

Then the mapping $L: V \to \GF$ defined by:

$\map L v := \innerprod v {v_0}$

is a bounded linear functional with norm equal to $\norm {v_0}$.


By Inner Product with Vector is Linear Functional, $L$ is a linear functional.

To check that $L$ is bounded:

\(\ds \cmod {\map L v}\) \(=\) \(\ds \cmod {\innerprod v {v_0} }\)
\(\ds \) \(\le\) \(\ds \norm v \norm {v_0}\) Cauchy-Bunyakovsky-Schwarz Inequality for Inner Product Spaces

Thus $L$ is bounded by $\norm {v_0}$.

Furthermore, note:

$\map L {v_0} = \innerprod {v_0} {v_0} = \norm {v_0}^2$

so that indeed $\norm {v_0}$ is the norm of $L$.