From 73445885d54edffcc9ae74525887b529a3f96165 Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Sat, 23 Mar 2024 18:08:44 +0100 Subject: Update --- .../inner-product-spaces.md | 177 --------------------- 1 file changed, 177 deletions(-) delete mode 100644 pages/functional-analysis-basics/inner-product-spaces.md (limited to 'pages/functional-analysis-basics/inner-product-spaces.md') 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 %} -- cgit v1.2.3-54-g00ecf