Inner Product with Vector is Bounded Linear Functional
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Theorem
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 $\norm {v_0}$.
Proof
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 $L$.
$\blacksquare$
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 3.$ The Riesz Representation Theorem