UpdateHEADmaster

4 files changed, 123 insertions, 32 deletions

diff --git a/pages/functional-analysis-basics/hilbert-spaces.md b/pages/functional-analysis-basics/hilbert-spaces.md
index a77e5c7..4cc46ec 100644
--- a/pages/functional-analysis-basics/hilbert-spaces.md
+++ b/pages/functional-analysis-basics/hilbert-spaces.md
@@ -2,10 +2,10 @@
 title: Hilbert Spaces
 parent: Functional Analysis Basics
 nav_order: 7
-published: false
 ---
 
 # {{ page.title }}
 
-{% proof %}
-{% endproof %}
+{% definition Hilbert Space %}
+A *Hilbert space* is a complete inner product space.
+{% enddefinition %}
diff --git a/pages/functional-analysis-basics/inner-product-spaces.md b/pages/functional-analysis-basics/inner-product-spaces/index.md
index 5f34a8a..56e31cf 100644
--- a/pages/functional-analysis-basics/inner-product-spaces.md
+++ b/pages/functional-analysis-basics/inner-product-spaces/index.md
@@ -1,13 +1,15 @@
 ---
 title: Inner Product Spaces
 parent: Functional Analysis Basics
-nav_order: 1
+nav_order: 6
+has_children: true
+has_toc: false
 ---
 
 # {{ page.title }}
 
 {% definition Inner Product Space %}
-An *inner product* on a real or complex vector space $X$
+An *inner product* (or *scalar product*) on a real or complex vector space $X$
 is a mapping
 
 $$
@@ -17,10 +19,21 @@ $$
 that is
 
 - linear in its second argument
+  
+  $$
+  \innerp{x}{y+z} = \innerp{x}{y} + \innerp{x}{z} \qquad
+  \innerp{x}{\alpha y} = \alpha \innerp{x}{y}
+  $$
+
 - conjugate symmetric
+  
+  $$
+  \overline{\innerp{x}{y}} = \innerp{x}{y}
+  $$
+
 - nondegenerate
 
-An *inner product space* is a pair $(X,\innerp{\cdot}{\cdot})$
+An *inner product space* (or *pre-Hilbert space*) is a pair $(X,\innerp{\cdot}{\cdot})$
 consisting of a real or complex vector space $X$
 and an inner product $\innerp{\cdot}{\cdot}$ on $X$.
 {% enddefinition %}
@@ -37,7 +50,7 @@ defines a norm on $X$.
 {% endproposition %}
 
 In this sense, every inner product space is also a normed space.
-As a consequence it is also a metric space and a topolgical space.
+As a consequence it is also a metric space and a topological space.
 
 The next theorem shows how the inner product can be recovered from the norm.
 
@@ -129,11 +142,26 @@ $$
 \norm{x+y}^2 + \norm{x-y}^2 = 2 \norm{x}^2 + 2 \norm{y}^2 \qquad \forall x,y \in X.
 $$
 
-In this case, the inner product is uniquely determined by $\norm{\cdot}$ and given by the polarization identity.
+In this case, the inner product is uniquely determined by $\norm{\cdot}$
+and given by the polarization identity.
+{% endtheorem %}
+
+{% theorem * Stewart’s Theorem %}
+Let $x$, $y$, $z$ be vectors of an inner product space.
+If $x$, $y$ and $z$ are colinear and $y$ lies inbetween $x$ and $y$,
+then we have
+
+$$
+\norm{p-x}^2 \norm{y-z} + \norm{p-z}^2 \norm{x-y} =
+\big\lparen \norm{p-y}^2 + \norm{x-y} \norm{y-z} \big\rparen \norm{x-z}
+$$
 {% endtheorem %}
 
+---
+
 {% theorem * Cauchy–Schwarz Inequality %}
-For all vectors $x$ and $y$ of an inner product space (with inner product $\innerp{\cdot}{\cdot}$ and induced norm $\norm{\cdot}$)
+For all vectors $x$ and $y$ of an inner product space
+(with inner product $\innerp{\cdot}{\cdot}$ and induced norm $\norm{\cdot}$)
 
 $$
 \abs{\innerp{x}{y}} \le \norm{x} \norm{y},
@@ -153,25 +181,5 @@ TODO
 {% endproof %}
 
 {% corollary Continuity of the Inner Product %}
-The inner product is jointly norm continous.
+The inner product is jointly norm continuous.
 {% endcorollary %}
-
-## Orthogonality
-
-{% definition Orthogonal Vectors %}
-Two vectors $x$ and $y$ of an inner product space $(X,\innerp{\cdot}{\cdot})$
-are said to be *orthogonal* or *perpendicular* if $\innerp{x}{y} = 0$,
-and this fact is indicated by writing $x \perp y$.
-{% enddefinition %}
-
-{% theorem * Pythagoras’ Theorem %}
-For all vectors $x$ and $y$ of an inner product space we have
-
-$$
-x \perp y \iff \norm{x+y}^2 = \norm{x}^2 + \norm{y}^2.
-$$
-{% endtheorem %}
-
-{% proof %}
-Immediate.
-{% endproof %}
diff --git a/pages/functional-analysis-basics/inner-product-spaces/orthogonality.md b/pages/functional-analysis-basics/inner-product-spaces/orthogonality.md
new file mode 100644
index 0000000..4e21138
--- /dev/null
+++ b/pages/functional-analysis-basics/inner-product-spaces/orthogonality.md
@@ -0,0 +1,79 @@
+---
+title: Orthogonality
+parent: Inner Product Spaces
+grand_parent: Functional Analysis Basics
+nav_order: 1
+---
+
+# {{ page.title }}
+
+{% definition Orthogonal Vectors %}
+Two vectors $x$ and $y$ of an inner product space $(X,\innerp{\cdot}{\cdot})$
+are said to be *orthogonal* or *perpendicular* if $\innerp{x}{y} = 0$,
+and this fact is indicated by writing $x \perp y$. \
+A set $A \subset X$ is called orthogonal,
+if the elements of $S$ are pairwise orthogonal to each other.
+{% enddefinition %}
+
+{% theorem * Pythagoras’ Theorem %}
+If $x$ and $y$ are orthogonal vectors of an inner product space, then
+
+$$
+\norm{x+y}^2 = \norm{x}^2 + \norm{y}^2.
+$$
+
+More generally,if $\braces{x_1,\ldots,x_n}$ is an orthogonal set in an inner product space, then
+
+$$
+\norm{x_1 + \cdots + x_n}^2 =
+\norm{x_1}^2 + \cdots + \norm{x_n}^2.
+$$
+{% endtheorem %}
+
+The converse of Pythagoras’ Theorem is true for real inner product space,
+but false in the complex case.
+For example, let $x$ be any unit vector in a complex inner product space.
+Then $x$ is not orthogonal to $ix$, since $\innerp{x}{ix} = i \ne 0$.
+However, $\norm{x+ix}^2 = \abs{1+i}^2 = 2 = 1 + 1 = \norm{x}^2 + \norm{ix}^2$.
+
+{% definition Orthonormal Set %}
+A subset $S$ of an inner product space is called *orthonormal*
+if we have for all $x,y \in S$
+
+$$
+\innerp{x}{y} = \begin{cases}
+0 & x=y, \\
+1 & x \ne y.
+\end{cases}
+$$
+{% enddefinition %}
+
+In other words, an orthonormal set is an orthogonal set of unit vectors.
+
+{% proposition %}
+Every orthonormal set is linearly independent.
+{% endproposition %}
+
+{% proof %}
+Suppose that $\braces{x_1,\ldots,x_n}$ is a finite subset of $S$ and that
+
+$$
+\alpha_1 x_1 + \cdots + \alpha_n x_n = 0
+$$
+
+for some scalars $\alpha_1,\ldots,\alpha_n$.
+Application of $\innerp{x_i}{\cdot}$ yields
+$\alpha_i = 0$ for all $i \in \braces{1,\ldots,n}$.
+{% endproof %}
+
+Recall that a subset $S$ of a normed space $X$ is called total
+if its span is dense in $X$.
+
+{% definition Orthonormal Basis %}
+A total orthonormal set in an inner product space is called
+*orthonormal basis* (or *complete orthonormal system*).
+{% enddefinition %}
+
+{% theorem %}
+Every Hilbert space has an orthonormal basis.
+{% endtheorem %}
diff --git a/pages/functional-analysis-basics/normed-spaces/index.md b/pages/functional-analysis-basics/normed-spaces.md
index c92d8c1..45afad1 100644
--- a/pages/functional-analysis-basics/normed-spaces/index.md
+++ b/pages/functional-analysis-basics/normed-spaces.md
@@ -2,8 +2,12 @@
 title: Normed Spaces
 parent: Functional Analysis Basics
 nav_order: 1
-has_children: true
-has_toc: false
 ---
 
 # {{ page.title }}
+
+{% theorem %}
+{% endtheorem %}
+
+{% proof %}
+{% endproof %}