Commit dd33ba7b by Michael Kohlhase

### adding inner product spaces and vievws

parent c3144597
 \begin{mhmodnl}[creators=miko]{inner-product-space}{en} \begin{definition} Let $F$ be the \trefi[field]{field} of \atrefii[realnumbers]{real}{real}{number} or \trefiis[complexnumbers]{complex}{numbers}, $V$ a \trefii[vector-space]{vector}{space} over $F$, and $\fun\innerproductOp{V,V}F$ a function with \begin{enumerate} \item $\innerproduct{x}y=\compconjugate{\innerproduct{y}x}$ (\defii{conjugate}{symmetry}) \item $\innerproduct{\smul{a}x}y=\comptimes{a{\innerproduct{x}y}}$ and $\innerproduct{\vadd{x,y}}z=\vadd{\innerproduct{x}z,\innerproduct{y}z}$ (\mtrefi[linear-map?linear]{linearity} in the first argument) \item $\realmethan{\innerproduct{x}x}0$ and $\eq{\innerproduct{x}x,0}$ iff $x=0$, (\defii{positive}{definiteness}), \end{enumerate} then $\mvstructure{V,\innerproductOp}$ is called an \defiii{inner}{product}{space} \end{definition} \end{mhmodnl} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
 \begin{modsig}[creators=miko]{inner-product-space} \gimport[smglom/arithmetics]{comparith} \gimport[smglom/linear-algebra]{linear-map} \symdef[name=inner-product]{innerproduct}[2]{\mixfixii[nobrackets]\langle{#1},{#2}\rangle} \symdef[name=inner-product]{innerproductOp}{\innerproduct\cdot\cdot} \symtest{innerproduct}{\innerproduct{x}y} \end{modsig} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
 \begin{mhmodnl}[creators=miko]{innerproduct-induced-norm}{en} \begin{definition} \inlineass[type=obligation,id=obl.norm-metric.en]{Let $\mvstructure{V,\innerproductOp}$ be a \trefiii[inner-product-space]{inner}{product}{space}, then $\fundefeq{x}{\anorm{x}}{\compsqrt{\innerproduct{x}x}}$ is a \trefi[norm]{norm}.} It is called the \adefii{norm induced}{induced}{norm} by $\innerproductOp$. \end{definition} \end{mhmodnl} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
 \begin{modsig}[creators=miko]{innerproduct-induced-norm} \gimport*{inner-product-space} \gimport{norm} \symii{induced}{norm} \end{modsig} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
 \begin{gviewnl}[creators=miko]{innerproduct-norm}{en} {norm}{innerproduct-induced-norm} Given a \trefiii[inner-product-space]{inner}{product}{space} with base set $V$ and \trefii[innerproduct-induced-norm]{induced}{norm} $\anormOp$, then $\mvstructure{V,\anormOp}$ is a \trefiii[norm]{normed}{vector}{space}. \end{gviewnl} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: % LocalWords: gve miko defeq trefii sproof spfstep
 \begin{gviewsig}[creators=miko]{innerproduct-norm}{norm}{innerproduct-induced-norm} \tassign{base-set}{base-set} \tassign{norm}{induced-norm} \end{gviewsig} \hypernym[by=norm-metric]{normed-vector-space}{inner-product-space} \hypernym[by=norm-metric]{norm}{inner-product} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
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