{"id":33436,"date":"2022-03-08T13:00:23","date_gmt":"2022-03-08T13:00:23","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33436"},"modified":"2025-07-23T10:48:18","modified_gmt":"2025-07-23T10:48:18","slug":"spatium-euclideum-rn","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/spatium-euclideum-rn\/","title":{"rendered":"Spatium Euclideum Rn"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>Spatium Euclideum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{\\mathbb{R}^n}<\/span><\/span><\/h1>\n<p style=\"text-align:center;\" dir=\"ltr\"><em>In hac lectione exploramus <strong>spatium euclideum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/strong>, eius structuram algebraicam et proprietates metricas. Disces de operationibus vectorialibus, <strong>producto interno<\/strong>, <strong>norma<\/strong> et <strong>distantia euclidea<\/strong>, conceptibus fundamentalibus in geometria et analysi. Expositionibus claris ac exemplis intuitivis, hic liber tibi facultatem dabit intellegendi quomodo spatium in multiplicibus dimensionibus mathematice exprimatur.<\/em><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\">\n<strong>Proposita Discendi:<\/strong><br \/>\nPost hanc lectionem, discipulus poterit:\n<\/p>\n<ol>\n<li><strong>Definire<\/strong> spatium euclideum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> eiusque proprietates fundamentales.<\/li>\n<li><strong>Explicare<\/strong> structuram vectorialem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> per eius operationes fundamentales.<\/li>\n<li><strong>Applicare<\/strong> productum internum ad angulos et projectiones inter vectores computandos.<\/li>\n<li><strong>Demonstrari<\/strong> proprietates algebraicas et metricas producti interni in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span>.<\/li>\n<li><strong>Adhibere<\/strong> normam euclideam ad magnitudinem vectoris determinandam.<\/li>\n<li><strong>Computare<\/strong> distantiam euclideam inter duo puncta in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> eiusque significationem geometricam enucleare.<\/li>\n<li><strong>Probare<\/strong> validitatem inaequalitatum fundamentalium, sicut Cauchy-Schwarz et inaequalitatis trianguli.<\/li>\n<\/ol>\n<p style=\"text-align:center;\" dir=\"ltr\"><strong>INDEX<\/strong><br \/>\n<a href=\"#1\">Spatium <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/a><br \/>\n<a href=\"#2\">Productum Internum<\/a><br \/>\n<a href=\"#3\">Norma et Distantia Euclidea<\/a><br \/>\n<a href=\"#4\">Conclusio<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/mV-G69l9LtI\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Spatium Vectoriale <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=mV-G69l9LtI&#038;t=123s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Certo antequam ad hunc locum perveneris, familiaris iam eras cum proprietatibus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R},<\/span><\/span> vel plani <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^2,<\/span><\/span> aut spatii <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^3.<\/span><\/span><\/span><\/strong><\/a> Hae omnes notiones utiles sunt ad intelligendum spatium <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n.<\/span><\/span> Imprimis, collectio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n = \\{\\vec{x} = (x_1, \\cdots, x_n) | x_1, \\cdots, x_n \\in \\mathbb{R}\\},<\/span><\/span> una cum operationibus solitis additionis vectorialis et multiplicationis per scalar, est spatium vectoriale. Huius rei indagationem incipiamus per considerationem operationum fundamentalium <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n.<\/span><\/span> <\/p>\n<h3>Operationes fundamentales <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=mV-G69l9LtI&#038;t=232s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1, \\cdots, x_n), \\vec{y}=(y_1, \\cdots, y_n)<\/span><\/span> vectores sunt in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span><\/span> est scalam realis quaelibet,<\/span><\/strong> <\/a>tum operationes <strong>additionis vectorum<\/strong> et <strong>multiplicationis per scalam<\/strong> sic describuntur:<\/p>\n<p><strong>Additio vectorum:<\/strong> Describitur per functionem sequentem:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rcrl} +:&amp; \\mathbb{R}^n \\times \\mathbb{R}^n &amp; \\longrightarrow &amp; \\mathbb{R}^n \\\\ &amp; (\\vec{x},\\vec{y}) &amp; \\longmapsto &amp; \\vec{x}+\\vec{y} = (x_1+y_1, \\cdots, x_n + y_n) \\end{array} <\/span><\/span><\/p>\n<p><strong>Multiplicatio per scalam:<\/strong> Describitur per functionem sequentem:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\begin{array}{rcrl} ():&amp; \\mathbb{R} \\times \\mathbb{R}^n &amp; \\longrightarrow &amp; \\mathbb{R}^n \\\\ &amp; (\\alpha,\\vec{x}) &amp; \\longmapsto &amp; (\\alpha\\vec{x}) = (\\alpha x_1, \\cdots, \\alpha x_n) \\end{array} <\/span>\n<h3>Proprietates spatii vectorialis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span><\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=mV-G69l9LtI&#038;t=428s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Spatium <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> instructum operationibus supra descriptis<\/span><\/strong><\/a> est <strong>spatium vectoriale,<\/strong> quia eius operationes additionis et multiplicationis per scalam satisfaciunt proprietatibus infra descriptis:<\/p>\n<p>Primum habemus proprietates <strong>commutativam<\/strong> et <strong>associativam.<\/strong><\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\vec{x} + \\vec{y} = \\vec{y} + \\vec{x}  \\\\ \\vec{x} + (\\vec{y}  + \\vec{z}) = (\\vec{x} + \\vec{y})  + \\vec{z}  \\\\ (\\alpha \\beta) \\vec{x}  = \\alpha (\\beta  \\vec{x}) = \\beta (\\alpha  \\vec{x}) = (\\beta\\alpha) \\vec{x}\n\n<\/span>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=\\displaystyle\\sum_{i=1}^n x_i\\hat{e}_i = x_1\\hat{e}_1 + \\cdots x_n\\hat{e}_n<\/span>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=\\displaystyle\\sum_{i=1}^n y_i\\hat{e}_i = y_1\\hat{e}_1 + \\cdots y_n\\hat{e}_n<\/span>\n<p>Hic explicite ostenditur coefficientes <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_i<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y_i<\/span><\/span> vectorum esse relativos ad basim spatii.<\/p>\n<h3>Conventio Summationis Einsteniana<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=518s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Conventio summationis Einsteniana<\/span><\/strong><\/a> sinit nos notationem vectorum in genere et producti interni in specie simpliciorem reddere. Si duas expressiones supra consideremus, videbimus indicem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">i<\/span><\/span> iterari tam in coefficientibus vectorum quam in elementis basis vectorialis; pro Einstenio, repetitio indicis sufficit ad intellegendam summam in expressione implicite contentam, ita scribi potest:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=  x_i\\hat{e}_i<\/span>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}= y_i\\hat{e}_i <\/span>\n<p>Hac notatione utens, productum internum exprimitur sic:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y} = x_i\\hat{e}_i \\cdot y_i\\hat{e}_i = x_iy_i \\underbrace{(\\hat{e}_i \\cdot \\hat{e}_i)}_{=1} = x_iy_i  <\/span>\n<p>In hac ultima aequatione assumptum est nos uti basi canonica.<\/p>\n<h3>Aliae Notationes pro Producto Interno<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=825s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Notatio vectorum earumque operationum non est eadem in omnibus contextibus,<\/span><\/strong><\/a> ea quam adhibui in primis paragraphis huius scripti est communissima in calculo. In algebra lineari autem, aliquando fit distinctio inter vectores et covectores:<\/p>\n<p>Quotiens loquimur de vectoribus, intelligimus quod dicitur \u00abvector columna\u00bb, qui repraesentatur matricialiter sic:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha^i = \\left( \\begin{array}{c}\\alpha_1 \\\\ \\vdots \\\\ \\alpha_n \\end{array} \\right)  <\/span>\n<p>At vero, cum loquimur de covectoribus, intelligimus \u00abvectorem ordinis\u00bb, qui sic repraesentatur matricialiter:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta_i = \\left( \\beta_1 \\; \\cdots \\; \\beta_n  \\right)  <\/span>\n<p>Ita, productum internum duorum vectorum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,\\cdots,x_n)<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,\\cdots,y_n)<\/span><\/span> interpretatur ut productum matriciale \u00abcovectoris\u00bb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x_i<\/span><\/span> cum vectore <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^i,<\/span><\/span> quod dat numerum realem sequentem:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left( x_1 \\; \\cdots \\; x_n  \\right) \\left( \\begin{array}{c}y_1 \\\\ \\vdots \\\\ y_n \\end{array} \\right)  = x_iy^i  <\/span>\n<p>Nota quod in hac ultima aequatione iterum apparet conventio Einsteniana: indices repetiti indicant summationem implicitam.<\/p>\n<p>Notatio quae sinit nos vectores et covectores distinguere per indices subscriptos et superscriptos appellatur \u00abnotatio covariantis\u00bb vel etiam \u00abnotatio tensorialis\u00bb, quae late adhibetur in theoria relativitatis specialis et generalis; haec notatio praeterea commodum habet laborandi cum tensoribus, conceptu qui generalizationem praebet super ea quae modo consideravimus, et quam in alia occasione accuratius tractabimus. In aliis disciplinis, ut in mechanica quantica, praeferitur notatio Bra\u2013Ket, ubi:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left&lt; x \\right| =\\left( x_1 \\; \\cdots \\; x_n  \\right) \\\\ \\\\ \\left|y\\right&gt; = \\left( \\begin{array}{c}y_1 \\\\ \\vdots \\\\ y_n \\end{array} \\right)\n\n <\/span>\n<p>Quo fit ut productum internum exprimatur in forma <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left&lt;x|y\\right&gt;.<\/span><\/span><\/p>\n<h3>Proprietates Producti Interni<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=HL85aSpHdsI&#038;t=1083s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Ex definitione producti interni elici possunt plures proprietates<\/span><\/strong><\/a> quae in futuris argumentis magni momenti erunt.<\/p>\n<p>Si productum internum adhibemus ad definire functionem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{\\omega}(\\vec{x})=\\vec{\\omega} \\cdot \\vec{x} = \\omega_i x^i,<\/span><\/span> tum videbimus functionem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{\\omega}<\/span><\/span> hoc modo definitam omnes proprietates functionum linearum possidere, cum facile demonstrari possit quod<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} \\tilde{\\omega}(\\alpha \\vec{x} + \\beta\\vec{y}) = \\alpha \\tilde{\\omega}(\\vec{x}) + \\beta\\tilde{\\omega}(\\vec{y}) \\end{array}<\/span>\n<p>et propterea obiecta ut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{\\omega}<\/span><\/span> quae ex producto interno definiuntur vocantur <strong>functionales lineares.<\/strong> Ut iam novimus, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}<\/span><\/span> est vector qui membrum est <strong>spatii vectorialis<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n,<\/span><\/span> et, ut in aliis contextibus patebit, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\tilde{\\omega}<\/span><\/span> est obiectum <strong>spatii dualis<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n.<\/span><\/span><\/p>\n<p>Ex hoc colligitur nexum arctissimum exsistere inter productum internum et functiones lineares; re vera, expressio quae omnes proprietates principales producti interni breviter enuntiat est: <em><strong>\u00abproductum internum est forma bilinearis, symmetrica, positiva et non-degenerata\u00bb.<\/strong><\/em> Videamus quid singulae partes huius enuntiationis significent:<\/p>\n<p>Quando dicimus <strong>productum internum esse formam bilinearem,<\/strong> significamus quod si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{z}<\/span><\/span> sunt vectores in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha,\\beta \\in \\mathbb{R},<\/span><\/span> tunc satisfiunt aequationes sequentes:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} \\vec{x}\\cdot(\\alpha \\vec{y} + \\beta\\vec{z}) = \\alpha (\\vec{x}\\cdot\\vec{y}) + \\beta(\\vec{x}\\cdot\\vec{z}) \\\\ \\\\ (\\alpha \\vec{x} + \\beta\\vec{y})\\cdot\\vec{z} = \\alpha (\\vec{x} \\cdot \\vec{z}) + \\beta(\\vec{y}\\cdot\\vec{z}) \\end{array}<\/span>\n<p>Productum internum <strong>symmetrica est forma<\/strong>, quia:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\forall(\\vec{x},\\vec{y}\\in\\mathbb{R}^n)(\\vec{x}\\cdot\\vec{y} = \\vec{y}\\cdot\\vec{x})<\/span>\n<p>Est etiam <strong>definite positiva<\/strong>, quia:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall\\vec{x}\\in\\mathbb{R}^n)(\\vec{x}\\cdot\\vec{x} \\geq 0)<\/span>\n<p>Et denique, est <strong>non-degenerata<\/strong>, quia:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{x} = 0 \\leftrightarrow \\vec{x}=\\vec{0}<\/span>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/vTFqDBEyU4Y\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Norma et Distantia Euclidea<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=174s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Norma est modus mensurandi magnitudinem vectoris,<\/span><\/strong><\/a> cum spatium vectoriale normam habeat, dicitur esse <strong>Spatium Vectoriale Normatum.<\/strong> Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lambda\\in\\mathbb{R},<\/span><\/span> tum functio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Norm( . )<\/span><\/span> norma est si condiciones sequentes implet: <\/p>\n<ol>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Norm(\\vec{x})\\geq 0<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Norm(\\vec{x}) = 0 \\leftrightarrow \\vec{x}=\\vec{0}<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Norm(\\lambda\\vec{x}) = |\\lambda| Norm(\\vec{x})<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Norm(\\vec{x} + \\vec{y}) \\leq Norm(\\vec{x}) + Norm(\\vec{y})<\/span><\/span><\/li>\n<\/ol>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=350s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Momentum notabile producti interni<\/span><\/strong><\/a> est quod hoc instrumentum praesertim utile est ad definiedum conceptum distantiae mathematico modo, qui congruit cum intuitiva nostra perceptione distantiae inter duo puncta. Pro omni <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\in\\mathbb{R}^n<\/span><\/span> definimus eius <strong>Normam Euclideam,<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\|<\/span><\/span> per aequationem:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x}\\| = \\sqrt{\\vec{x}\\cdot\\vec{x}}<\/span>\n<p>Ex hoc dicimus <strong>normam euclideam esse normam a producto interno inductam.<\/strong><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=846s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Distantia, sive metrica,<\/span><\/strong><\/a> est functio quae indicat \u00abquantum duo elementa coniuncti separantur\u00bb. Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}, \\vec{y}, \\vec{z}\\in\\mathbb{R}^n<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lambda\\in\\mathbb{R},<\/span><\/span> tunc functio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist( . )<\/span><\/span> est distantia si proprietates sequentes satisfacit:<\/p>\n<ol>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist(\\vec{x},\\vec{y})=0 \\leftrightarrow \\vec{x}=\\vec{y}<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist(\\vec{x},\\vec{y})=Dist(\\vec{y},\\vec{x})\\geq 0<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist(\\vec{x},\\vec{z})\\leq Dist(\\vec{x},\\vec{y}) + Dist(\\vec{y},\\vec{z})<\/span><\/span><\/li>\n<\/ol>\n<p>Ultima expressio appellatur <strong>Inaequalitas Triangularis,<\/strong> et si haec non impleatur, functio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Dist(.)<\/span><\/span> vocatur \u00abpseudo-distantia\u00bb sive \u00abpseudo-metrica\u00bb. Spatium Vectoriale quod distantia instructum est vocatur <strong>Spatium Metricum.<\/strong><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=1013s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Ex Norma Euclidea<\/span><\/strong><\/a> definitur <strong>Distantia Euclidea<\/strong> inter duos vectores. Si habemus duos vectores <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n,<\/span><\/span> tunc distantia euclidea inter hos duos vectores, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dist_e(\\vec{x},\\vec{y})<\/span><\/span> definita est per:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dist_e(\\vec{x},\\vec{y}) = \\|\\vec{x} - \\vec{y}\\|<\/span>\n<p>Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}=(x_1,\\cdots,x_n)<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{y}=(y_1,\\cdots, y_n),<\/span><\/span> tunc facile demonstrari potest ex proprietatibus producti interni et normae quod:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dist_e(\\vec{x},\\vec{y}) = \\sqrt{\\displaystyle \\sum_{i=1}^n (x_i - y_i)^2}<\/span>\n<p>Si spatium vectoriale <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span> cum distantia euclidea instruitur, quod obtinetur vocatur <strong>Spatium Euclideum.<\/strong><\/p>\n<p>Ex hoc dicitur <strong>metricam spatii euclidei esse metricam a norma euclidea inductam.<\/strong><\/p>\n<h3>Proprietates Normae Euclideae<\/h3>\n<p><\/strong> Cum nostra inquisitio proprie versetur in Spatio Euclideo, expedit proprietates normae euclideae recognoscere.<\/p>\n<h4>Inaequalitas Cauchy\u2013Schwarz<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=1624s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n,<\/span><\/span><\/span><\/strong><\/a> tunc sequens proprietas valet:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">|\\vec{x}\\cdot\\vec{y}|\\leq \\|\\vec{x}\\|\\|\\vec{y}\\|<\/span>\n<p>DEMONSTRATIO:<\/p>\n<p>Sit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lambda = (\\vec{x}\\cdot\\vec{y})\/\\|\\vec{y}\\|^2,<\/span><\/span> tunc habetur:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} 0\\leq \\|\\vec{x} - \\lambda \\vec{y}\\|^2 &amp;= (\\vec{x} - \\lambda\\vec{y}) \\cdot (\\vec{x} - \\lambda\\vec{y}) \\\\ \\\\\n\n\\displaystyle &amp;= \\vec{x}\\cdot\\vec{x} - \\lambda\\vec{x}\\cdot\\vec{y} + \\lambda\\vec{y}\\cdot\\vec{x} + \\lambda^2(\\vec{y}\\cdot\\vec{y})\\\\ \\\\\n\n&amp;= \\|\\vec{x}\\|^2 - 2\\lambda(\\vec{x}\\cdot\\vec{y}) + \\lambda^2 \\|\\vec{y}\\|^2 \\\\ \\\\\n\n\\displaystyle &amp;= \\|\\vec{x}\\|^2 - 2\\left(\\frac{\\vec{x}\\cdot\\vec{y}}{\\|\\vec{y}\\|^2}\\right)(\\vec{x}\\cdot\\vec{y}) + \\left(\\frac{\\vec{x}\\cdot\\vec{y}}{{\\|\\vec{y}\\|^2}}\\right)^2 {\\|\\vec{y}\\|^2}\\\\ \\\\\n\n\\displaystyle &amp;= \\|\\vec{x}\\|^2 - 2\\left(\\frac{(\\vec{x}\\cdot\\vec{y})^2}{\\|\\vec{y}\\|^2}\\right) + \\frac{\\left(\\vec{x}\\cdot\\vec{y}\\right)^2}{\\|\\vec{y}\\|^2}\\\\ \\\\\n\n&amp;= \\|\\vec{x}\\|^2 - \\frac{\\left(\\vec{x}\\cdot\\vec{y}\\right)^2}{\\|\\vec{y}\\|^2} \\end{array}<\/span>\n<p>Quare possumus dicere: <\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle 0 \\leq \\|\\vec{x}\\|^2 - \\frac{\\left(\\vec{x}\\cdot\\vec{y}\\right)^2}{\\|\\vec{y}\\|^2} <\/span>\n<p>Et propterea: <\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\left(\\vec{x}\\cdot\\vec{y}\\right)^2 \\leq \\|\\vec{x}\\|^2 \\|\\vec{y}\\|^2 <\/span>\n<p>Denique, radicem extrahentes, pervenitur ad propositum: <\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> |\\vec{x}\\cdot\\vec{y}| \\leq \\|\\vec{x}\\| \\|\\vec{y}\\|<\/span> \u2b1b<\/p>\n<h4>Inaequalitas Triangularis<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=vTFqDBEyU4Y&#038;t=2065s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #ff0000;\">Sint <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x},\\vec{y}\\in\\mathbb{R}^n,<\/span><\/span><\/span><\/strong><\/a> hi vectores relationem satisfaciunt:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} + \\vec{y}\\| \\leq \\|\\vec{x}\\| + \\|\\vec{y}\\|<\/span>\n<p>DEMONSTRATIO:<\/p>\n<p>Primum animadvertamus:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl} \\|\\vec{x} + \\vec{y}\\|^2 &amp;= (\\vec{x} + \\vec{y})\\cdot(\\vec{x} + \\vec{y}) \\\\ \\\\\n\n&amp;=\\|\\vec{x}\\|^2 + 2(\\vec{x}\\cdot\\vec{y}) + \\|\\vec{y}\\|^2 \\end{array}<\/span>\n<p>Cum sint verae relationes:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{x}\\cdot\\vec{y}\\leq |\\vec{x}\\cdot\\vec{y}| \\leq \\|\\vec{x}\\|\\vec{y}\\|<\/span>\n<p>Possumus scribere sequentia:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\|\\vec{x} + \\vec{y}\\|^2 &amp;\\leq  \\|x\\|^2 + 2\\|\\vec{x}\\|\\vec{y}\\| + \\|\\vec{y}\\|^2 \\\\ \\\\\n\n&amp;\\leq  \\left(\\|\\vec{x}\\|  + \\|\\vec{y}\\| \\right)^2\n\n\\end{array}<\/span>\n<p>Denique, radicem extrahendo, pervenitur ad id quod demonstrandum erat:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{x} + \\vec{y}\\|\\leq  \\|\\vec{x}\\|  + \\|\\vec{y}\\|<\/span> \u2b1b <\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Conclusio<\/h2>\n<p>Per totam hanc lectionem exploravimus proprietates fundamentales spatii euclidei <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span>, tractantes eius structuras algebraicas et metricas. Coepimus definiendo eius operationes fundamentales, ut sunt additio vectorum et productum internum, ita constituentes eius naturam ut spatii vectorialis. Deinde, altius perscrutati sumus conceptum producti interni eiusque momentum in geometria <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span><\/span>, praesertim eius interpretationem matricialem et relationem ad functiones lineares illustrantes.<\/p>\n<p>Postea, examinavimus normam euclideam et distantiam ex ea inductam, ostendentes quomodo hae notiones nos sinant longitudines et distantias in hoc spatio quantificare. Praeterea, recensuimus proprietates fundamentales, sicut inaequalitatem Cauchy\u2013Schwarz:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> |\\vec{x}\\cdot\\vec{y}| \\leq \\|\\vec{x}\\| \\|\\vec{y}\\| <\/span>\n<p>et inaequalitatem triangularem:<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\|\\vec{x} + \\vec{y}\\|\\leq  \\|\\vec{x}\\|  + \\|\\vec{y}\\| <\/span>\n<p>quae sunt notiones clavis ad progressum theoriarum altiorum in analysi et geometria.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Spatium Euclideum In hac lectione exploramus spatium euclideum , eius structuram algebraicam et proprietates metricas. Disces de operationibus vectorialibus, producto interno, norma et distantia euclidea, conceptibus fundamentalibus in geometria et analysi. Expositionibus claris ac exemplis intuitivis, hic liber tibi facultatem dabit intellegendi quomodo spatium in multiplicibus dimensionibus mathematice exprimatur. Proposita Discendi: Post hanc lectionem, discipulus [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":32541,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":3,"footnotes":""},"categories":[1316,1340,1298],"tags":[],"class_list":["post-33436","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-linearis","category-calculus-multivariabilis","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Spatium Euclideum Rn - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Explora proprietates Spatii Euclidei Rn , eius structuram algebraicam, productum internum, normam atque distantiam euclideam in multiplicibus dimensionibus.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/la\/spatium-euclideum-rn\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Spatium Euclideum Rn\" \/>\n<meta property=\"og:description\" content=\"Explora proprietates Spatii Euclidei Rn , eius structuram algebraicam, productum internum, normam atque distantiam euclideam in multiplicibus dimensionibus.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/spatium-euclideum-rn\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2022-03-08T13:00:23+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-07-23T10:48:18+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/03\/vectores-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Spatium Euclideum Rn\" \/>\n<meta name=\"twitter:description\" content=\"Explora proprietates Spatii Euclidei Rn , eius structuram algebraicam, productum internum, normam atque distantiam euclideam in multiplicibus dimensionibus.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2022\/03\/vectores.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/spatium-euclideum-rn\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/spatium-euclideum-rn\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Spatium Euclideum Rn\",\"datePublished\":\"2022-03-08T13:00:23+00:00\",\"dateModified\":\"2025-07-23T10:48:18+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/spatium-euclideum-rn\\\/\"},\"wordCount\":2150,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/spatium-euclideum-rn\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2022\\\/03\\\/vectores.jpg\",\"articleSection\":[\"Algebra Linearis\",\"Calculus Multivariabilis\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/spatium-euclideum-rn\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/spatium-euclideum-rn\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/spatium-euclideum-rn\\\/\",\"name\":\"Spatium Euclideum Rn - 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