Editing Riesz representation theorem (section)

=== Observations ===

If <math>\varphi \in H^*</math> then 
<math display=block>\varphi \left(f_{\varphi}\right) = \left\langle f_{\varphi}, f_{\varphi} \right\rangle = \left\|f_{\varphi}\right\|^2 = \|\varphi\|^2.</math> 
So in particular, <math>\varphi \left(f_{\varphi}\right) \geq 0</math> is always real and furthermore, <math>\varphi \left(f_{\varphi}\right) = 0</math> if and only if <math>f_{\varphi} = 0</math> if and only if <math>\varphi = 0.</math> 

'''Linear functionals as affine hyperplanes'''

A non-trivial continuous linear functional <math>\varphi</math> is often interpreted geometrically by identifying it with the affine hyperplane <math>A := \varphi^{-1}(1)</math> (the kernel <math>\ker\varphi = \varphi^{-1}(0)</math> is also often visualized alongside <math>A := \varphi^{-1}(1)</math> although knowing <math>A</math> is enough to reconstruct <math>\ker \varphi</math> because if <math>A = \varnothing</math> then <math>\ker \varphi = H</math> and otherwise <math>\ker \varphi = A - A</math>). In particular, the norm of <math>\varphi</math> should somehow be interpretable as the "norm of the hyperplane <math>A</math>". When <math>\varphi \neq 0</math> then the Riesz representation theorem provides such an interpretation of <math>\|\varphi\|</math> in terms of the affine hyperplane<ref group=note name="VectorSpaceStructureOnAffineHyperplanesInducedByDualSpace" /> 
<math>A := \varphi^{-1}(1)</math> as follows: using the notation from the theorem's statement, from <math>\|\varphi\|^2 \neq 0</math> it follows that <math>C := \varphi^{-1}\left(\|\varphi\|^2\right) = \|\varphi\|^2 \varphi^{-1}(1) = \|\varphi\|^2 A</math> and so <math>\|\varphi\| = \left\|f_{\varphi}\right\| = \inf_{c \in C} \|c\|</math> implies <math>\|\varphi\| = \inf_{a \in A} \|\varphi\|^2 \|a\|</math> and thus <math>\|\varphi\| = \frac{1}{\inf_{a \in A} \|a\|}.</math> 
This can also be seen by applying the [[Hilbert projection theorem]] to <math>A</math> and concluding that the global minimum point of the map <math>A \to [0, \infty)</math> defined by <math>a \mapsto \|a\|</math> is <math>\frac{f_{\varphi}}{\|\varphi\|^2} \in A.</math> 
The formulas 
<math display=block>\frac{1}{\inf_{a \in A} \|a\|} = \sup_{a \in A} \frac{1}{\|a\|}</math>
provide the promised interpretation of the linear functional's norm <math>\|\varphi\|</math> entirely in terms of its associated affine hyperplane <math>A = \varphi^{-1}(1)</math> (because with this formula, knowing only the {{em|set}} <math>A</math> is enough to describe the norm of its associated linear {{em|functional}}). Defining <math>\frac{1}{\infty} := 0,</math> the [[infimum]] formula 
<math display=block>\|\varphi\| = \frac{1}{\inf_{a \in \varphi^{-1}(1)} \|a\|}</math>
will also hold when <math>\varphi = 0.</math> 
When the supremum is taken in <math>\R</math> (as is typically assumed), then the supremum of the empty set is <math>\sup \varnothing = - \infty</math> but if the supremum is taken in the non-negative reals <math>[0, \infty)</math> (which is the [[Image of a function|image]]/range of the norm <math>\|\,\cdot\,\|</math> when <math>\dim H > 0</math>) then this supremum is instead <math>\sup \varnothing = 0,</math> in which case the supremum formula <math>\|\varphi\| = \sup_{a \in \varphi^{-1}(1)} \frac{1}{\|a\|}</math> will also hold when <math>\varphi = 0</math> (although the atypical equality <math>\sup \varnothing = 0</math> is usually unexpected and so risks causing confusion).