Inner Product Linear Functional Representations
Why
We can identify any linear functional $F: \R ^n \to \R $ with a vector $y \in \R ^n$ so that $F(x) = \ip{x,y}$. We generalize this result to complete inner product spaces.
Motivating result
The following is known as the Riesz representation theorem (or Riesz-Fréchet representation theorem, or Riesz theorem, or Riesz-Fréchet theorem).
Let $((V, k), \ip{\cdot ,\cdot })$ be a complete
inner product space and let $F: V \to k$ be a
continuous linear functional on $V$.
There exists a unique $y \in V$ so that
\[ F(x) = \ip{x, y} \]
for all $x \in V$. Moreover $\norm{y} = \dnorm{F}$.Clearly $\R ^n$ is a complete inner product space, and so this this theorem says the expected. We can identify linear functionals on $\R ^n$ with elements (vectors) in $\R ^n$.1
- Future editions will expand further. ↩︎