Inner Product with Vector is 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 linear functional.
Proof
Let us directly check the definition of linear functional:
\(\ds \map L {\alpha v + \beta w}\) | \(=\) | \(\ds \innerprod { \alpha v + \beta w } {v_0}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \innerprod {\alpha v} {v_0} + \innerprod {\beta w} {v_0}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \alpha \innerprod v {v_0} + \beta \innerprod w {w_0}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \alpha \map L v + \beta \map L w\) |
$\blacksquare$
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 3.$ The Riesz Representation Theorem