From a1b5de688d879069b5e1192057d71572c7bc5368 Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Thu, 29 Feb 2024 17:32:24 +0100 Subject: Update --- .../cauchys-integral-formula.md | 8 +- .../one-complex-variable/cauchys-theorem.md | 4 +- .../inner-product-spaces.md | 177 +++++++++++++++++++++ pages/general-topology/metric-spaces/index.md | 10 +- .../lebesgue-integral/convergence-theorems.md | 6 +- .../lebesgue-integral/the-lp-spaces.md | 22 +++ .../measure-theory/borels-sets.md | 2 +- .../measure-theory/measurable-maps.md | 2 +- pages/operator-algebras/banach-algebras/index.md | 2 +- 9 files changed, 220 insertions(+), 13 deletions(-) create mode 100644 pages/functional-analysis-basics/inner-product-spaces.md (limited to 'pages') diff --git a/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md b/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md index 6ac0803..f7414d5 100644 --- a/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md +++ b/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md @@ -20,7 +20,7 @@ $$ {% proof %} {% endproof %} -{% theorem * Cauchy's Integral Formula (Generalization) %} +{% theorem * Cauchy’s Integral Formula (Generalization) %} Let $f : G \to \CC$ be a function holomorphic in an open subset $G \subset \CC$. Then the $n$th derivative $f^{(n)}$ exists for every $n \in \NN$. If $\gamma$ is a contour in $G$ such that the interior of $\gamma$ is contained in $G$, @@ -44,7 +44,7 @@ and is often used to compute the integral. ## Many Consequences -{% theorem * Cauchy's Estimate %} +{% theorem * Cauchy’s Estimate %} Let $f$ be holomorphic on an open set containing the disc with center $a$ and radius $r>0$. Then @@ -74,7 +74,7 @@ and the right-hand side of the inequality reduces to the desired expression. Recall that an *entire* function is a holomorphic function that is defined everywhere in the complex plane. -{% theorem * Liouville's Theorem %} +{% theorem * Liouville’s Theorem %} Every bounded entire function is constant. {% endtheorem %} @@ -82,7 +82,7 @@ Every bounded entire function is constant. Consider an entire function $f$ and assume that $\norm{f(z)} \le M$ for all $z \in \CC$ and some $M > 0$. Since $f$ is holomorphic on the whole plane, we may make -[Cauchy's Estimate](#cauchy-s-estimate) +[Cauchy’s Estimate](#cauchy-s-estimate) for all disks centered at any point $a \in \CC$ and with any radius $r>0$. For the first derivative, we have $\norm{f'(a)} \le M/r$, which tends to $0$ for $r \to \infty$. Hence $f' = 0$ in the whole plane. This diff --git a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md index 6d78e89..eb040ca 100644 --- a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md +++ b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md @@ -7,7 +7,7 @@ nav_order: 2 # {{ page.title }} -{% theorem Cauchy's Theorem (Homotopy Version) %} +{% theorem Cauchy’s Theorem (Homotopy Version) %} Let $G$ be a connected open subset of the complex plane. Let $f : G \to \CC$ be a holomorphic function. If $\gamma_0$, $\gamma_1$ are homotopic closed curves in $G$, then @@ -29,7 +29,7 @@ $$ {{ page.title }} has a converse: -{% theorem * Morera's Theorem %} +{% theorem * Morera’s Theorem %} Let $G \subset \CC$ be open and let $f : G \to \CC$ be a continuous function. If $\int_{\gamma} f(z) \, dz = 0$ for every contour $\gamma$ contained in $G$, then $f$ is holomorphic in $G$. diff --git a/pages/functional-analysis-basics/inner-product-spaces.md b/pages/functional-analysis-basics/inner-product-spaces.md new file mode 100644 index 0000000..5f34a8a --- /dev/null +++ b/pages/functional-analysis-basics/inner-product-spaces.md @@ -0,0 +1,177 @@ +--- +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/general-topology/metric-spaces/index.md b/pages/general-topology/metric-spaces/index.md index 52b2b4c..6aef400 100644 --- a/pages/general-topology/metric-spaces/index.md +++ b/pages/general-topology/metric-spaces/index.md @@ -60,8 +60,16 @@ Let $(X,d)$ be a (semi-)metric space. The proofs are straightforward. +{% definition Isometry %} +Suppose $(X,d_X)$ and $(Y,d_Y)$ are metric spaces. +We say that a mapping $f : X \to Y$ is *isometric* or an *isometry* if it obeys +$d_Y \big\lparen f(x),f(x') \big\rparen = d_X(x,x')$ for all $x,x' \in X$. +{% enddefinition %} + +As a consequence of **(M1)**, +every isometry is injective. + TODO -- isometry - metric induced by a norm - metric product diff --git a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md index 1a34820..6808280 100644 --- a/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md +++ b/pages/measure-and-integration/lebesgue-integral/convergence-theorems.md @@ -4,9 +4,9 @@ parent: Lebesgue Integral grand_parent: Measure and Integration nav_order: 2 description: > -We state and prove the most important convergence theorems of Lebesgue -integration theory such as the Monotone Convergence Theorem, Fatou’s Lemma, and the -Dominated Convergence Theorem. + We state and prove the most important convergence theorems of Lebesgue + integration theory such as the Monotone Convergence Theorem, Fatou’s Lemma, and the + Dominated Convergence Theorem. --- # {{ page.title }} diff --git a/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md index 8482e87..0424117 100644 --- a/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md +++ b/pages/measure-and-integration/lebesgue-integral/the-lp-spaces.md @@ -23,6 +23,28 @@ Endowed with pointwise addition and scalar multiplication $\mathscr{L}^p(X,\mathcal{A},\mu)$ becomes a vector space. {% endproposition %} +{% proof %} +We show that $\mathscr{L}^p := \mathscr{L}^p(X,\mathcal{A},\mu)$ is a linear subspace of +the vector space of all $\KK$-valued functions on $X$. +The set $\mathscr{L}^p$ is nonempty since +it contains the zero function. +Now, suppose $f$ and $g$ are in $\mathscr{L}^p$. +Then the sum $f+g$ is measurable, because $f$ and $g$ are measurable. +Moreover, the function $\abs{f+g}^p$ is integrable, because we have the estimate + +$$ +\abs{f+g}^p +\le (\abs{f} + \abs{g})^p +\le \big\lparen 2 \max(\abs{f},\abs{g}) \big\rparen^p +\le 2^p (\abs{f}^p + \abs{g}^p), +$$ + +where $\abs{f}^p$ and $\abs{g}^p$ are integrable. +This proves that $f+g$ lies in $\mathscr{L}^p$. +Finally, it is easy to see that $\alpha f$ lies in $\mathscr{L}^p$ +for any scalar $\alpha \in \KK$. +{% endproof %} + {% proposition %} $\norm{\cdot}_p$ is a seminorm on $\mathscr{L}^p(X,\mathcal{A},\mu)$. {% endproposition %} diff --git a/pages/measure-and-integration/measure-theory/borels-sets.md b/pages/measure-and-integration/measure-theory/borels-sets.md index 737a7c8..0cdb142 100644 --- a/pages/measure-and-integration/measure-theory/borels-sets.md +++ b/pages/measure-and-integration/measure-theory/borels-sets.md @@ -2,7 +2,7 @@ title: Borel Sets parent: Measure Theory grand_parent: Measure and Integration -nav_order: 2 +nav_order: 3 --- # {{ page.title }} diff --git a/pages/measure-and-integration/measure-theory/measurable-maps.md b/pages/measure-and-integration/measure-theory/measurable-maps.md index 5b7a76e..dc9f4d7 100644 --- a/pages/measure-and-integration/measure-theory/measurable-maps.md +++ b/pages/measure-and-integration/measure-theory/measurable-maps.md @@ -2,7 +2,7 @@ title: Measurable Maps parent: Measure Theory grand_parent: Measure and Integration -nav_order: 3 +nav_order: 2 --- # {{ page.title }} diff --git a/pages/operator-algebras/banach-algebras/index.md b/pages/operator-algebras/banach-algebras/index.md index 3a427a7..5f33816 100644 --- a/pages/operator-algebras/banach-algebras/index.md +++ b/pages/operator-algebras/banach-algebras/index.md @@ -188,7 +188,7 @@ $$ $$ This shows that $R$ is a bounded entire function. Now -[Liouville's Theorem](/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.html#liouvilles-theorem) +[Liouville’s Theorem](/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.html#liouvilles-theorem) (for vector-valued functions) implies that $R$ is constant. This is contradictory because XXX {% endproof %} -- cgit v1.2.3-54-g00ecf