From 73445885d54edffcc9ae74525887b529a3f96165 Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Sat, 23 Mar 2024 18:08:44 +0100 Subject: Update --- pages/functional-analysis-basics/hilbert-spaces.md | 6 +- .../inner-product-spaces.md | 177 -------------------- .../inner-product-spaces/index.md | 185 +++++++++++++++++++++ .../inner-product-spaces/orthogonality.md | 79 +++++++++ pages/functional-analysis-basics/normed-spaces.md | 13 ++ .../normed-spaces/index.md | 9 - .../measure-theory/measures.md | 4 +- 7 files changed, 282 insertions(+), 191 deletions(-) delete mode 100644 pages/functional-analysis-basics/inner-product-spaces.md create mode 100644 pages/functional-analysis-basics/inner-product-spaces/index.md create mode 100644 pages/functional-analysis-basics/inner-product-spaces/orthogonality.md create mode 100644 pages/functional-analysis-basics/normed-spaces.md delete mode 100644 pages/functional-analysis-basics/normed-spaces/index.md (limited to 'pages') 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.md deleted file mode 100644 index 5f34a8a..0000000 --- a/pages/functional-analysis-basics/inner-product-spaces.md +++ /dev/null @@ -1,177 +0,0 @@ ---- -title: Inner Product Spaces -parent: Functional Analysis Basics -nav_order: 1 ---- - -# {{ page.title }} - -{% definition Inner Product Space %} -An *inner product* on a real or complex vector space $X$ -is a mapping - -$$ -\innerp{\cdot}{\cdot} : X \times X \to \KK -$$ - -that is - -- linear in its second argument -- conjugate symmetric -- nondegenerate - -An *inner product 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 %} - -{% proposition Norm Induced by an Inner Product %} -If $\innerp{\cdot}{\cdot}$ is an inner product -on a real or complex vector space $X$, then - -$$ -\norm{x} = \sqrt{\innerp{x}{x}} \qquad \forall x \in X -$$ - -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. - -The next theorem shows how the inner product can be recovered from the norm. - -{% theorem * Polarization Identity %} -For all vectors $x$ and $y$ of a real inner product space - -$$ -4 \innerp{x}{y} = \norm{x+y}^2 - \norm{x-y}^2. -$$ - -For all vectors $x$ and $y$ of a complex inner product space - -$$ -4 \innerp{x}{y} = \norm{x+y}^2 - \norm{x-y}^2 + i \norm{x-iy}^2 - i \norm{x+iy}^2. -$$ -{% endtheorem %} - -Note that the complex polarization identity takes the slightly different form - -$$ -4 \innerp{x}{y} = \norm{x+y}^2 - \norm{x-y}^2 + i \norm{x\mathrel{\color{red}+}iy}^2 - i \norm{x\mathrel{\color{red}-}iy}^2, -$$ - -if we follow the convention that the inner product is conjugate linear in its second argument. - -{% proof %} -In the real case, the inner product is symmetric, and we have - -$$ -\norm{x \pm y}^2 = \norm{x}^2 \pm 2 \innerp{x}{y} + \norm{y}^2 -$$ - -for all vectors $x$ and $y$. -Taking the difference yields the desired result. - -In the complex case, the inner product is conjugate symmetric, and we have - -$$ -\norm{x \pm y}^2 = \norm{x}^2 \pm 2 \Re \innerp{x}{y} + \norm{y}^2 -$$ - -for all vectors $x$ and $y$. This implies - -$$ -\begin{aligned} -\norm{x + \phantom{i}y}^2 - \norm{x - \phantom{i}y}^2 &= 4 \Re \innerp{x}{y}, \\ -\norm{x - iy}^2 - \norm{x + iy}^2 &= 4 \Im \innerp{x}{y}. -\end{aligned} -$$ - -The second equation follows from the first by -substituting $y$ with $-iy$ and -using that $\Re \innerp{x}{-iy} = \Re (-i\innerp{x}{y}) = \Im \innerp{x}{y}$. -To obtain the polarization Identity, multiply the second equation with $i$ and then add it to the first. -{% endproof %} - -{% theorem * General Polarization Identity %} -Let $X$ be a complex inner product space. -Let $\zeta$ be a $n$-th root of unity with $\zeta \ne 1$ and $\zeta^2 \ne 1$. -Then - -$$ -\innerp{x}{y} = \frac{1}{n} \sum_{k=0}^{n-1} \zeta^k \norm{x + \zeta^k y}^2 \qquad \forall x,y \in X. -$$ -{% endtheorem %} - -As a special case, for $\zeta = i$ and $n=4$, we obtain - -$$ -\innerp{x}{y} = \frac{1}{4} \sum_{k=0}^{3} i^k \norm{x + i^k y}^2. -$$ - -{% proof %} -TODO -{% endproof %} - -For an arbitrary normed space, -the polarization identity does not, in general, -define an inner product. -The following theorem, gives a condition for when it does. - -{% theorem * Parallelogram Law %} -Let $X$ be a real or complex normed space. -A norm $\norm{\cdot}$ on $X$ is induced by -an inner product $\innerp{\cdot}{\cdot}$ on $X$, -if and only if $\norm{\cdot}$ satisfies the *parallelogram law* - -$$ -\norm{x+y}^2 + \norm{x-y}^2 = 2 \norm{x}^2 + 2 \norm{y}^2 \qquad \forall x,y \in X. -$$ - -In this case, the inner product is uniquely determined by $\norm{\cdot}$ and given by the polarization identity. -{% 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 $\norm{\cdot}$) - -$$ -\abs{\innerp{x}{y}} \le \norm{x} \norm{y}, -$$ - -and equality holds precisely when $x$ and $y$ are linearly dependent. -{% endtheorem %} - -Expressed only in terms of the inner product, the Cauchy–Schwarz Inequality reads - -$$ -\abs{\innerp{x}{y}}^2 \le \innerp{x}{x} \innerp{y}{y}. -$$ - -{% proof %} -TODO -{% endproof %} - -{% corollary Continuity of the Inner Product %} -The inner product is jointly norm continous. -{% 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/index.md b/pages/functional-analysis-basics/inner-product-spaces/index.md new file mode 100644 index 0000000..56e31cf --- /dev/null +++ b/pages/functional-analysis-basics/inner-product-spaces/index.md @@ -0,0 +1,185 @@ +--- +title: Inner Product Spaces +parent: Functional Analysis Basics +nav_order: 6 +has_children: true +has_toc: false +--- + +# {{ page.title }} + +{% definition Inner Product Space %} +An *inner product* (or *scalar product*) on a real or complex vector space $X$ +is a mapping + +$$ +\innerp{\cdot}{\cdot} : X \times X \to \KK +$$ + +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* (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 %} + +{% proposition Norm Induced by an Inner Product %} +If $\innerp{\cdot}{\cdot}$ is an inner product +on a real or complex vector space $X$, then + +$$ +\norm{x} = \sqrt{\innerp{x}{x}} \qquad \forall x \in X +$$ + +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 topological space. + +The next theorem shows how the inner product can be recovered from the norm. + +{% theorem * Polarization Identity %} +For all vectors $x$ and $y$ of a real inner product space + +$$ +4 \innerp{x}{y} = \norm{x+y}^2 - \norm{x-y}^2. +$$ + +For all vectors $x$ and $y$ of a complex inner product space + +$$ +4 \innerp{x}{y} = \norm{x+y}^2 - \norm{x-y}^2 + i \norm{x-iy}^2 - i \norm{x+iy}^2. +$$ +{% endtheorem %} + +Note that the complex polarization identity takes the slightly different form + +$$ +4 \innerp{x}{y} = \norm{x+y}^2 - \norm{x-y}^2 + i \norm{x\mathrel{\color{red}+}iy}^2 - i \norm{x\mathrel{\color{red}-}iy}^2, +$$ + +if we follow the convention that the inner product is conjugate linear in its second argument. + +{% proof %} +In the real case, the inner product is symmetric, and we have + +$$ +\norm{x \pm y}^2 = \norm{x}^2 \pm 2 \innerp{x}{y} + \norm{y}^2 +$$ + +for all vectors $x$ and $y$. +Taking the difference yields the desired result. + +In the complex case, the inner product is conjugate symmetric, and we have + +$$ +\norm{x \pm y}^2 = \norm{x}^2 \pm 2 \Re \innerp{x}{y} + \norm{y}^2 +$$ + +for all vectors $x$ and $y$. This implies + +$$ +\begin{aligned} +\norm{x + \phantom{i}y}^2 - \norm{x - \phantom{i}y}^2 &= 4 \Re \innerp{x}{y}, \\ +\norm{x - iy}^2 - \norm{x + iy}^2 &= 4 \Im \innerp{x}{y}. +\end{aligned} +$$ + +The second equation follows from the first by +substituting $y$ with $-iy$ and +using that $\Re \innerp{x}{-iy} = \Re (-i\innerp{x}{y}) = \Im \innerp{x}{y}$. +To obtain the polarization Identity, multiply the second equation with $i$ and then add it to the first. +{% endproof %} + +{% theorem * General Polarization Identity %} +Let $X$ be a complex inner product space. +Let $\zeta$ be a $n$-th root of unity with $\zeta \ne 1$ and $\zeta^2 \ne 1$. +Then + +$$ +\innerp{x}{y} = \frac{1}{n} \sum_{k=0}^{n-1} \zeta^k \norm{x + \zeta^k y}^2 \qquad \forall x,y \in X. +$$ +{% endtheorem %} + +As a special case, for $\zeta = i$ and $n=4$, we obtain + +$$ +\innerp{x}{y} = \frac{1}{4} \sum_{k=0}^{3} i^k \norm{x + i^k y}^2. +$$ + +{% proof %} +TODO +{% endproof %} + +For an arbitrary normed space, +the polarization identity does not, in general, +define an inner product. +The following theorem, gives a condition for when it does. + +{% theorem * Parallelogram Law %} +Let $X$ be a real or complex normed space. +A norm $\norm{\cdot}$ on $X$ is induced by +an inner product $\innerp{\cdot}{\cdot}$ on $X$, +if and only if $\norm{\cdot}$ satisfies the *parallelogram law* + +$$ +\norm{x+y}^2 + \norm{x-y}^2 = 2 \norm{x}^2 + 2 \norm{y}^2 \qquad \forall x,y \in X. +$$ + +In this case, the inner product is uniquely determined by $\norm{\cdot}$ +and given by the polarization identity. +{% 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 $\norm{\cdot}$) + +$$ +\abs{\innerp{x}{y}} \le \norm{x} \norm{y}, +$$ + +and equality holds precisely when $x$ and $y$ are linearly dependent. +{% endtheorem %} + +Expressed only in terms of the inner product, the Cauchy–Schwarz Inequality reads + +$$ +\abs{\innerp{x}{y}}^2 \le \innerp{x}{x} \innerp{y}{y}. +$$ + +{% proof %} +TODO +{% endproof %} + +{% corollary Continuity of the Inner Product %} +The inner product is jointly norm continuous. +{% endcorollary %} 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.md b/pages/functional-analysis-basics/normed-spaces.md new file mode 100644 index 0000000..45afad1 --- /dev/null +++ b/pages/functional-analysis-basics/normed-spaces.md @@ -0,0 +1,13 @@ +--- +title: Normed Spaces +parent: Functional Analysis Basics +nav_order: 1 +--- + +# {{ page.title }} + +{% theorem %} +{% endtheorem %} + +{% proof %} +{% endproof %} diff --git a/pages/functional-analysis-basics/normed-spaces/index.md b/pages/functional-analysis-basics/normed-spaces/index.md deleted file mode 100644 index c92d8c1..0000000 --- a/pages/functional-analysis-basics/normed-spaces/index.md +++ /dev/null @@ -1,9 +0,0 @@ ---- -title: Normed Spaces -parent: Functional Analysis Basics -nav_order: 1 -has_children: true -has_toc: false ---- - -# {{ page.title }} diff --git a/pages/measure-and-integration/measure-theory/measures.md b/pages/measure-and-integration/measure-theory/measures.md index c843881..5ad3308 100644 --- a/pages/measure-and-integration/measure-theory/measures.md +++ b/pages/measure-and-integration/measure-theory/measures.md @@ -7,7 +7,7 @@ nav_order: 4 # {{ page.title }} -{% definition %} +{% definition Measure %} A *measure* on a σ-algebra $\mathcal{A}$ on a set $X$ is mapping $\mu : \mathcal{A} \to [0,\infty]$ such that @@ -17,7 +17,7 @@ is mapping $\mu : \mathcal{A} \to [0,\infty]$ such that $$ \mu \bigg\lparen \bigcup_{n=1}^{\infty} A_n \! \bigg\rparen - = \sum_{n=0}^{\infty} \mu(A_n). + = \sum_{n=1}^{\infty} \mu(A_n). $$ {% enddefinition %} -- cgit v1.2.3-54-g00ecf