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author | Justin Gassner <justin.gassner@mailbox.org> | 2024-03-23 18:08:44 +0100 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-03-23 18:08:44 +0100 |
commit | 73445885d54edffcc9ae74525887b529a3f96165 (patch) | |
tree | 525c54151d604a4cf2f69601fc88f45e8673adc2 /pages/functional-analysis-basics | |
parent | a1b5de688d879069b5e1192057d71572c7bc5368 (diff) | |
download | site-73445885d54edffcc9ae74525887b529a3f96165.tar.zst |
Diffstat (limited to 'pages/functional-analysis-basics')
-rw-r--r-- | pages/functional-analysis-basics/hilbert-spaces.md | 6 | ||||
-rw-r--r-- | pages/functional-analysis-basics/inner-product-spaces/index.md (renamed from pages/functional-analysis-basics/inner-product-spaces.md) | 62 | ||||
-rw-r--r-- | pages/functional-analysis-basics/inner-product-spaces/orthogonality.md | 79 | ||||
-rw-r--r-- | pages/functional-analysis-basics/normed-spaces.md (renamed from pages/functional-analysis-basics/normed-spaces/index.md) | 8 |
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 %} |