Formal definition of euclidean space












1












$begingroup$


Is there an agreed upon, precise definition of the euclidean space? Although I have searched in may places, I can't find anything that is rigorous. From what I have read, I have thought of defining it like this:




Definition: The euclidean space of $n$ dimensions, $E^n$, is defined as the topology generated by the basis ($R^{n},d$), where $R^{n}$ is the set (Not the cartesian product of the standard real line topology) and $d$ is the Euclidean metric $d(x,y) = Sigma^{n}_{i=1}(x^{2}_{i}-y^{2}_{i})$ (where $x = (x_{1},dots ,x_{n})$ and $y=((y_{1},dots ,y_{n})$).




Would the definition above be accurate?



Similarly, would it be accurate to define the $n$ sphere, $S^{n}$ (as a topological space) as the subset topology of ${p in R^{n} | d(x,p) = a}$ (Where $a in R^{+}$ and $x in R^{n}$) inherited from the euclidean topology?






metric-spaces definition geometric-topology







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    The product topology and metric topology are the same, in this case.
    $endgroup$
    – Cameron Buie
    Jan 5 at 15:09










  • $begingroup$
    All topologies generated by a norm on a finite-dimensional space over the real or complex numbers are in fact the same.
    $endgroup$
    – ncmathsadist
    Jan 5 at 15:11










  • $begingroup$
    So can I use the definition I have provided? (Sorry for the bluntness)
    $endgroup$
    – Aryaman Gupta
    Jan 5 at 15:14










  • $begingroup$
    I suspect that the short answer to your question is "no". There have been several similar questions before, including What is the difference between a Hilbert space and Euclidean space?, What really is the modern definition of Euclidean spaces?, Is the Euclidean plane equal to $mathbb{R}^2$?. Personally I remain confused, so I'm just commenting, not answering!
    $endgroup$
    – Calum Gilhooley
    Jan 5 at 15:50






  • 1




    $begingroup$
    Following up on the comment of CalumGilhooley, this question invites ambiguous answers because it does not address the following key issue: What are the salient mathematical structures of Euclidean space? Simply its points, as the answer of @ItsJustAMeasureBro suggests? Its points together with its lines? Its points together with its metric, as the definition in your question suggests? Its points, lines, metric, angles, and angle measurement? All of the above together with its inner product, as one link in Calum's comment suggests? .........
    $endgroup$
    – Lee Mosher
    Jan 5 at 18:54


















1












$begingroup$


Is there an agreed upon, precise definition of the euclidean space? Although I have searched in may places, I can't find anything that is rigorous. From what I have read, I have thought of defining it like this:




Definition: The euclidean space of $n$ dimensions, $E^n$, is defined as the topology generated by the basis ($R^{n},d$), where $R^{n}$ is the set (Not the cartesian product of the standard real line topology) and $d$ is the Euclidean metric $d(x,y) = Sigma^{n}_{i=1}(x^{2}_{i}-y^{2}_{i})$ (where $x = (x_{1},dots ,x_{n})$ and $y=((y_{1},dots ,y_{n})$).




Would the definition above be accurate?



Similarly, would it be accurate to define the $n$ sphere, $S^{n}$ (as a topological space) as the subset topology of ${p in R^{n} | d(x,p) = a}$ (Where $a in R^{+}$ and $x in R^{n}$) inherited from the euclidean topology?






metric-spaces definition geometric-topology







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    The product topology and metric topology are the same, in this case.
    $endgroup$
    – Cameron Buie
    Jan 5 at 15:09










  • $begingroup$
    All topologies generated by a norm on a finite-dimensional space over the real or complex numbers are in fact the same.
    $endgroup$
    – ncmathsadist
    Jan 5 at 15:11










  • $begingroup$
    So can I use the definition I have provided? (Sorry for the bluntness)
    $endgroup$
    – Aryaman Gupta
    Jan 5 at 15:14










  • $begingroup$
    I suspect that the short answer to your question is "no". There have been several similar questions before, including What is the difference between a Hilbert space and Euclidean space?, What really is the modern definition of Euclidean spaces?, Is the Euclidean plane equal to $mathbb{R}^2$?. Personally I remain confused, so I'm just commenting, not answering!
    $endgroup$
    – Calum Gilhooley
    Jan 5 at 15:50






  • 1




    $begingroup$
    Following up on the comment of CalumGilhooley, this question invites ambiguous answers because it does not address the following key issue: What are the salient mathematical structures of Euclidean space? Simply its points, as the answer of @ItsJustAMeasureBro suggests? Its points together with its lines? Its points together with its metric, as the definition in your question suggests? Its points, lines, metric, angles, and angle measurement? All of the above together with its inner product, as one link in Calum's comment suggests? .........
    $endgroup$
    – Lee Mosher
    Jan 5 at 18:54
















1












1








1


1



$begingroup$


Is there an agreed upon, precise definition of the euclidean space? Although I have searched in may places, I can't find anything that is rigorous. From what I have read, I have thought of defining it like this:




Definition: The euclidean space of $n$ dimensions, $E^n$, is defined as the topology generated by the basis ($R^{n},d$), where $R^{n}$ is the set (Not the cartesian product of the standard real line topology) and $d$ is the Euclidean metric $d(x,y) = Sigma^{n}_{i=1}(x^{2}_{i}-y^{2}_{i})$ (where $x = (x_{1},dots ,x_{n})$ and $y=((y_{1},dots ,y_{n})$).




Would the definition above be accurate?



Similarly, would it be accurate to define the $n$ sphere, $S^{n}$ (as a topological space) as the subset topology of ${p in R^{n} | d(x,p) = a}$ (Where $a in R^{+}$ and $x in R^{n}$) inherited from the euclidean topology?






metric-spaces definition geometric-topology







share|cite|improve this question











$endgroup$




Is there an agreed upon, precise definition of the euclidean space? Although I have searched in may places, I can't find anything that is rigorous. From what I have read, I have thought of defining it like this:




Definition: The euclidean space of $n$ dimensions, $E^n$, is defined as the topology generated by the basis ($R^{n},d$), where $R^{n}$ is the set (Not the cartesian product of the standard real line topology) and $d$ is the Euclidean metric $d(x,y) = Sigma^{n}_{i=1}(x^{2}_{i}-y^{2}_{i})$ (where $x = (x_{1},dots ,x_{n})$ and $y=((y_{1},dots ,y_{n})$).




Would the definition above be accurate?



Similarly, would it be accurate to define the $n$ sphere, $S^{n}$ (as a topological space) as the subset topology of ${p in R^{n} | d(x,p) = a}$ (Where $a in R^{+}$ and $x in R^{n}$) inherited from the euclidean topology?






metric-spaces definition geometric-topology




metric-spaces definition geometric-topology






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 5 at 15:08







Aryaman Gupta

















asked Jan 5 at 14:59









Aryaman GuptaAryaman Gupta

507




507








  • 1




    $begingroup$
    The product topology and metric topology are the same, in this case.
    $endgroup$
    – Cameron Buie
    Jan 5 at 15:09










  • $begingroup$
    All topologies generated by a norm on a finite-dimensional space over the real or complex numbers are in fact the same.
    $endgroup$
    – ncmathsadist
    Jan 5 at 15:11










  • $begingroup$
    So can I use the definition I have provided? (Sorry for the bluntness)
    $endgroup$
    – Aryaman Gupta
    Jan 5 at 15:14










  • $begingroup$
    I suspect that the short answer to your question is "no". There have been several similar questions before, including What is the difference between a Hilbert space and Euclidean space?, What really is the modern definition of Euclidean spaces?, Is the Euclidean plane equal to $mathbb{R}^2$?. Personally I remain confused, so I'm just commenting, not answering!
    $endgroup$
    – Calum Gilhooley
    Jan 5 at 15:50






  • 1




    $begingroup$
    Following up on the comment of CalumGilhooley, this question invites ambiguous answers because it does not address the following key issue: What are the salient mathematical structures of Euclidean space? Simply its points, as the answer of @ItsJustAMeasureBro suggests? Its points together with its lines? Its points together with its metric, as the definition in your question suggests? Its points, lines, metric, angles, and angle measurement? All of the above together with its inner product, as one link in Calum's comment suggests? .........
    $endgroup$
    – Lee Mosher
    Jan 5 at 18:54
















  • 1




    $begingroup$
    The product topology and metric topology are the same, in this case.
    $endgroup$
    – Cameron Buie
    Jan 5 at 15:09










  • $begingroup$
    All topologies generated by a norm on a finite-dimensional space over the real or complex numbers are in fact the same.
    $endgroup$
    – ncmathsadist
    Jan 5 at 15:11










  • $begingroup$
    So can I use the definition I have provided? (Sorry for the bluntness)
    $endgroup$
    – Aryaman Gupta
    Jan 5 at 15:14










  • $begingroup$
    I suspect that the short answer to your question is "no". There have been several similar questions before, including What is the difference between a Hilbert space and Euclidean space?, What really is the modern definition of Euclidean spaces?, Is the Euclidean plane equal to $mathbb{R}^2$?. Personally I remain confused, so I'm just commenting, not answering!
    $endgroup$
    – Calum Gilhooley
    Jan 5 at 15:50






  • 1




    $begingroup$
    Following up on the comment of CalumGilhooley, this question invites ambiguous answers because it does not address the following key issue: What are the salient mathematical structures of Euclidean space? Simply its points, as the answer of @ItsJustAMeasureBro suggests? Its points together with its lines? Its points together with its metric, as the definition in your question suggests? Its points, lines, metric, angles, and angle measurement? All of the above together with its inner product, as one link in Calum's comment suggests? .........
    $endgroup$
    – Lee Mosher
    Jan 5 at 18:54










1




1




$begingroup$
The product topology and metric topology are the same, in this case.
$endgroup$
– Cameron Buie
Jan 5 at 15:09




$begingroup$
The product topology and metric topology are the same, in this case.
$endgroup$
– Cameron Buie
Jan 5 at 15:09












$begingroup$
All topologies generated by a norm on a finite-dimensional space over the real or complex numbers are in fact the same.
$endgroup$
– ncmathsadist
Jan 5 at 15:11




$begingroup$
All topologies generated by a norm on a finite-dimensional space over the real or complex numbers are in fact the same.
$endgroup$
– ncmathsadist
Jan 5 at 15:11












$begingroup$
So can I use the definition I have provided? (Sorry for the bluntness)
$endgroup$
– Aryaman Gupta
Jan 5 at 15:14




$begingroup$
So can I use the definition I have provided? (Sorry for the bluntness)
$endgroup$
– Aryaman Gupta
Jan 5 at 15:14












$begingroup$
I suspect that the short answer to your question is "no". There have been several similar questions before, including What is the difference between a Hilbert space and Euclidean space?, What really is the modern definition of Euclidean spaces?, Is the Euclidean plane equal to $mathbb{R}^2$?. Personally I remain confused, so I'm just commenting, not answering!
$endgroup$
– Calum Gilhooley
Jan 5 at 15:50




$begingroup$
I suspect that the short answer to your question is "no". There have been several similar questions before, including What is the difference between a Hilbert space and Euclidean space?, What really is the modern definition of Euclidean spaces?, Is the Euclidean plane equal to $mathbb{R}^2$?. Personally I remain confused, so I'm just commenting, not answering!
$endgroup$
– Calum Gilhooley
Jan 5 at 15:50




1




1




$begingroup$
Following up on the comment of CalumGilhooley, this question invites ambiguous answers because it does not address the following key issue: What are the salient mathematical structures of Euclidean space? Simply its points, as the answer of @ItsJustAMeasureBro suggests? Its points together with its lines? Its points together with its metric, as the definition in your question suggests? Its points, lines, metric, angles, and angle measurement? All of the above together with its inner product, as one link in Calum's comment suggests? .........
$endgroup$
– Lee Mosher
Jan 5 at 18:54






$begingroup$
Following up on the comment of CalumGilhooley, this question invites ambiguous answers because it does not address the following key issue: What are the salient mathematical structures of Euclidean space? Simply its points, as the answer of @ItsJustAMeasureBro suggests? Its points together with its lines? Its points together with its metric, as the definition in your question suggests? Its points, lines, metric, angles, and angle measurement? All of the above together with its inner product, as one link in Calum's comment suggests? .........
$endgroup$
– Lee Mosher
Jan 5 at 18:54












2 Answers
2






active

oldest

votes


















0












$begingroup$

The formal definition is probably something close to this.




Euclidean $n$-space, sometimes called Cartesian space or simply $n$-space, is the space of all $n$-tuples of real numbers, ($x_1, x_2, ..., x_n$). Such $n$-tuples are sometimes called points, although other nomenclature may be used (see below). The totality of $n$-space is commonly denoted $mathbb R^n$, although older literature uses the symbol $mathbb E^n$ (or actually, its non-doublestruck variant $E^n$; O'Neill 1966, p. 3).







share|cite|improve this answer











$endgroup$













  • $begingroup$
    Depending on the context, Euclidean Space usually doesn't fix an origin and so there is a distinction between points and translations between them. $mathbb R^n$ can be used to stand for both the points and the translations, but that's sort of incidental. As an aside, I wouldn't personally trust mathworld for definitions; I've found pages with multiple inconsistent definitions, pages with definitions that seem to agree with no other internet source, etc.
    $endgroup$
    – Mark S.
    Jan 8 at 11:03





















1












$begingroup$

Ray Bowen, in his Introduction to Vectors and Tensors, Vol 2, section 43, defines a "Euclidean Point Space":




Consider an inner product space $V$ and a set $E$. The set $E$ is a
Euclidean point space if there exists a function $fcolon E times E to V$ such that:



(a) $f(x, y) = f(x, z) + f(z, y)$, for $x, y, zin E$ and



(b) For every $xin E$ and $vin V$ there exists a unique element $yin E$ such that
$f(x, y) = v$.



The elements of $E$ are called points, and the inner product space $V$ is called the translation space.
We say that $f(x, y)$ is the vector determined by the end point $x$ and the initial point $y$.







share|cite|improve this answer











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    2 Answers
    2






    active

    oldest

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    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    0












    $begingroup$

    The formal definition is probably something close to this.




    Euclidean $n$-space, sometimes called Cartesian space or simply $n$-space, is the space of all $n$-tuples of real numbers, ($x_1, x_2, ..., x_n$). Such $n$-tuples are sometimes called points, although other nomenclature may be used (see below). The totality of $n$-space is commonly denoted $mathbb R^n$, although older literature uses the symbol $mathbb E^n$ (or actually, its non-doublestruck variant $E^n$; O'Neill 1966, p. 3).







    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      Depending on the context, Euclidean Space usually doesn't fix an origin and so there is a distinction between points and translations between them. $mathbb R^n$ can be used to stand for both the points and the translations, but that's sort of incidental. As an aside, I wouldn't personally trust mathworld for definitions; I've found pages with multiple inconsistent definitions, pages with definitions that seem to agree with no other internet source, etc.
      $endgroup$
      – Mark S.
      Jan 8 at 11:03


















    0












    $begingroup$

    The formal definition is probably something close to this.




    Euclidean $n$-space, sometimes called Cartesian space or simply $n$-space, is the space of all $n$-tuples of real numbers, ($x_1, x_2, ..., x_n$). Such $n$-tuples are sometimes called points, although other nomenclature may be used (see below). The totality of $n$-space is commonly denoted $mathbb R^n$, although older literature uses the symbol $mathbb E^n$ (or actually, its non-doublestruck variant $E^n$; O'Neill 1966, p. 3).







    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      Depending on the context, Euclidean Space usually doesn't fix an origin and so there is a distinction between points and translations between them. $mathbb R^n$ can be used to stand for both the points and the translations, but that's sort of incidental. As an aside, I wouldn't personally trust mathworld for definitions; I've found pages with multiple inconsistent definitions, pages with definitions that seem to agree with no other internet source, etc.
      $endgroup$
      – Mark S.
      Jan 8 at 11:03
















    0












    0








    0





    $begingroup$

    The formal definition is probably something close to this.




    Euclidean $n$-space, sometimes called Cartesian space or simply $n$-space, is the space of all $n$-tuples of real numbers, ($x_1, x_2, ..., x_n$). Such $n$-tuples are sometimes called points, although other nomenclature may be used (see below). The totality of $n$-space is commonly denoted $mathbb R^n$, although older literature uses the symbol $mathbb E^n$ (or actually, its non-doublestruck variant $E^n$; O'Neill 1966, p. 3).







    share|cite|improve this answer











    $endgroup$



    The formal definition is probably something close to this.




    Euclidean $n$-space, sometimes called Cartesian space or simply $n$-space, is the space of all $n$-tuples of real numbers, ($x_1, x_2, ..., x_n$). Such $n$-tuples are sometimes called points, although other nomenclature may be used (see below). The totality of $n$-space is commonly denoted $mathbb R^n$, although older literature uses the symbol $mathbb E^n$ (or actually, its non-doublestruck variant $E^n$; O'Neill 1966, p. 3).








    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Jan 5 at 15:34









    EdOverflow

    25519




    25519










    answered Jan 5 at 15:09









    ItsJustAMeasureBroItsJustAMeasureBro

    342




    342












    • $begingroup$
      Depending on the context, Euclidean Space usually doesn't fix an origin and so there is a distinction between points and translations between them. $mathbb R^n$ can be used to stand for both the points and the translations, but that's sort of incidental. As an aside, I wouldn't personally trust mathworld for definitions; I've found pages with multiple inconsistent definitions, pages with definitions that seem to agree with no other internet source, etc.
      $endgroup$
      – Mark S.
      Jan 8 at 11:03




















    • $begingroup$
      Depending on the context, Euclidean Space usually doesn't fix an origin and so there is a distinction between points and translations between them. $mathbb R^n$ can be used to stand for both the points and the translations, but that's sort of incidental. As an aside, I wouldn't personally trust mathworld for definitions; I've found pages with multiple inconsistent definitions, pages with definitions that seem to agree with no other internet source, etc.
      $endgroup$
      – Mark S.
      Jan 8 at 11:03


















    $begingroup$
    Depending on the context, Euclidean Space usually doesn't fix an origin and so there is a distinction between points and translations between them. $mathbb R^n$ can be used to stand for both the points and the translations, but that's sort of incidental. As an aside, I wouldn't personally trust mathworld for definitions; I've found pages with multiple inconsistent definitions, pages with definitions that seem to agree with no other internet source, etc.
    $endgroup$
    – Mark S.
    Jan 8 at 11:03






    $begingroup$
    Depending on the context, Euclidean Space usually doesn't fix an origin and so there is a distinction between points and translations between them. $mathbb R^n$ can be used to stand for both the points and the translations, but that's sort of incidental. As an aside, I wouldn't personally trust mathworld for definitions; I've found pages with multiple inconsistent definitions, pages with definitions that seem to agree with no other internet source, etc.
    $endgroup$
    – Mark S.
    Jan 8 at 11:03













    1












    $begingroup$

    Ray Bowen, in his Introduction to Vectors and Tensors, Vol 2, section 43, defines a "Euclidean Point Space":




    Consider an inner product space $V$ and a set $E$. The set $E$ is a
    Euclidean point space if there exists a function $fcolon E times E to V$ such that:



    (a) $f(x, y) = f(x, z) + f(z, y)$, for $x, y, zin E$ and



    (b) For every $xin E$ and $vin V$ there exists a unique element $yin E$ such that
    $f(x, y) = v$.



    The elements of $E$ are called points, and the inner product space $V$ is called the translation space.
    We say that $f(x, y)$ is the vector determined by the end point $x$ and the initial point $y$.







    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      Ray Bowen, in his Introduction to Vectors and Tensors, Vol 2, section 43, defines a "Euclidean Point Space":




      Consider an inner product space $V$ and a set $E$. The set $E$ is a
      Euclidean point space if there exists a function $fcolon E times E to V$ such that:



      (a) $f(x, y) = f(x, z) + f(z, y)$, for $x, y, zin E$ and



      (b) For every $xin E$ and $vin V$ there exists a unique element $yin E$ such that
      $f(x, y) = v$.



      The elements of $E$ are called points, and the inner product space $V$ is called the translation space.
      We say that $f(x, y)$ is the vector determined by the end point $x$ and the initial point $y$.







      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        Ray Bowen, in his Introduction to Vectors and Tensors, Vol 2, section 43, defines a "Euclidean Point Space":




        Consider an inner product space $V$ and a set $E$. The set $E$ is a
        Euclidean point space if there exists a function $fcolon E times E to V$ such that:



        (a) $f(x, y) = f(x, z) + f(z, y)$, for $x, y, zin E$ and



        (b) For every $xin E$ and $vin V$ there exists a unique element $yin E$ such that
        $f(x, y) = v$.



        The elements of $E$ are called points, and the inner product space $V$ is called the translation space.
        We say that $f(x, y)$ is the vector determined by the end point $x$ and the initial point $y$.







        share|cite|improve this answer











        $endgroup$



        Ray Bowen, in his Introduction to Vectors and Tensors, Vol 2, section 43, defines a "Euclidean Point Space":




        Consider an inner product space $V$ and a set $E$. The set $E$ is a
        Euclidean point space if there exists a function $fcolon E times E to V$ such that:



        (a) $f(x, y) = f(x, z) + f(z, y)$, for $x, y, zin E$ and



        (b) For every $xin E$ and $vin V$ there exists a unique element $yin E$ such that
        $f(x, y) = v$.



        The elements of $E$ are called points, and the inner product space $V$ is called the translation space.
        We say that $f(x, y)$ is the vector determined by the end point $x$ and the initial point $y$.








        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 8 at 9:19









        Babelfish

        1,187520




        1,187520










        answered Jan 5 at 17:01









        user21793user21793

        1592




        1592






























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            Probability when a professor distributes a quiz and homework assignment to a class of n students.

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            0 0 $begingroup$ Need help with this problem. Suppose our lazy professor collects a quiz and a homework assignment from a class of n students one day, then distributes both the quizzes and the homework assignments back to the class in a random fashion for grading. Each student receives one quiz and one homework assignment to grade. (a) What is the probability that every student receives someone else's quiz to grade, and someone else's homework to grade? (b) What is the probability that no student receives both their own quiz and their own homework assignment to grade? In this case, some students may receive their own quiz, and others may receive their own homework assignment. (c) Compute the limiting probability as n approaches infinity in each case. ...
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            Aardman Animations

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            Aardman Animations, Ltd. Aardman Animations Tipo privada Atividade Animação em stop-motion , Animação em CGI Fundação 1972 Fundador(es) Peter Lord David Sproxton Sede Bristol, Inglaterra  Reino Unido Proprietário(s) Dreamworks Pessoas-chave Peter Lord David Sproxton Nick Park Divisões Aardman Features Aardman Digital Aardman Commercials Aardman Broadcast Aardman International Aardman Rights Aardman Effects Aardman 3-D Systems Aardman Nathan Love AardBoiled Website oficial aardman.com ...
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            Are they similar matrix

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            0 $begingroup$ Do $begin{bmatrix} 0&i&0\0&0&1\0&0&0 end{bmatrix} $ and $begin{bmatrix} 0&0&0\-i&0&0\0&1&0 end{bmatrix} $ are similar.Is this True/false Clearly both are nilpotent and one is conjucate transpose of other but how to know if they are similar.i'm stuck. Please help me linear-algebra vector-spaces eigenvalues-eigenvectors generalizedeigenvector share | cite | improve this question edited Dec 10 '18 at 6:38 Vasanth Kris asked Dec 10 '18 at 6:19 ...
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