Formal definition of euclidean space
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
asked 2 days ago
Aryaman GuptaAryaman Gupta
335
add a comment |
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
asked 2 days ago
Aryaman GuptaAryaman Gupta
335
1
The product topology and metric topology are the same, in this case.
– Cameron Buie
2 days ago
All topologies generated by a norm on a finite-dimensional space over the real or complex numbers are in fact the same.
– ncmathsadist
2 days ago
So can I use the definition I have provided? (Sorry for the bluntness)
– Aryaman Gupta
2 days ago
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!
– Calum Gilhooley
2 days ago
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? .........
– Lee Mosher
2 days ago
add a comment |
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
asked 2 days ago
Aryaman GuptaAryaman Gupta
335
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
asked 2 days ago
Aryaman GuptaAryaman Gupta
335
asked 2 days ago
Aryaman GuptaAryaman Gupta
335
edited 2 days ago
Aryaman Gupta
asked 2 days ago
Aryaman GuptaAryaman Gupta
335
asked 2 days ago
Aryaman GuptaAryaman Gupta
335
asked 2 days ago
Aryaman GuptaAryaman Gupta
335
335
1
The product topology and metric topology are the same, in this case.
– Cameron Buie
2 days ago
All topologies generated by a norm on a finite-dimensional space over the real or complex numbers are in fact the same.
– ncmathsadist
2 days ago
So can I use the definition I have provided? (Sorry for the bluntness)
– Aryaman Gupta
2 days ago
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!
– Calum Gilhooley
2 days ago
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? .........
– Lee Mosher
2 days ago
add a comment |
1
The product topology and metric topology are the same, in this case.
– Cameron Buie
2 days ago
All topologies generated by a norm on a finite-dimensional space over the real or complex numbers are in fact the same.
– ncmathsadist
2 days ago
So can I use the definition I have provided? (Sorry for the bluntness)
– Aryaman Gupta
2 days ago
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!
– Calum Gilhooley
2 days ago
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? .........
– Lee Mosher
2 days ago
1
1
The product topology and metric topology are the same, in this case.
– Cameron Buie
2 days ago
The product topology and metric topology are the same, in this case.
– Cameron Buie
2 days ago
All topologies generated by a norm on a finite-dimensional space over the real or complex numbers are in fact the same.
– ncmathsadist
2 days ago
All topologies generated by a norm on a finite-dimensional space over the real or complex numbers are in fact the same.
– ncmathsadist
2 days ago
So can I use the definition I have provided? (Sorry for the bluntness)
– Aryaman Gupta
2 days ago
So can I use the definition I have provided? (Sorry for the bluntness)
– Aryaman Gupta
2 days ago
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!
– Calum Gilhooley
2 days ago
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!
– Calum Gilhooley
2 days ago
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? .........
– Lee Mosher
2 days ago
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? .........
– Lee Mosher
2 days ago
add a comment |
2 Answers
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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).
edited 2 days ago
EdOverflow
1958
answered 2 days ago
ItsJustAMeasureBroItsJustAMeasureBro
362
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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 f: E × E →V such that:
(a) f(x, y) = f(x, z) + f(z, y), x, y, z∈E and
(b) For every x∈E and v∈V there exists a unique element y∈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.
answered 2 days ago
user21793user21793
1492
add a comment |
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2 Answers
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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).
edited 2 days ago
EdOverflow
1958
answered 2 days ago
ItsJustAMeasureBroItsJustAMeasureBro
362
New contributor
ItsJustAMeasureBro is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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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).
edited 2 days ago
EdOverflow
1958
answered 2 days ago
ItsJustAMeasureBroItsJustAMeasureBro
362
New contributor
ItsJustAMeasureBro is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
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).
edited 2 days ago
EdOverflow
1958
answered 2 days ago
ItsJustAMeasureBroItsJustAMeasureBro
362
New contributor
ItsJustAMeasureBro is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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).
edited 2 days ago
EdOverflow
1958
answered 2 days ago
ItsJustAMeasureBroItsJustAMeasureBro
362
New contributor
ItsJustAMeasureBro is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 2 days ago
EdOverflow
1958
edited 2 days ago
EdOverflow
1958
edited 2 days ago
EdOverflow
1958
1958
answered 2 days ago
ItsJustAMeasureBroItsJustAMeasureBro
362
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ItsJustAMeasureBro is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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answered 2 days ago
ItsJustAMeasureBroItsJustAMeasureBro
362
answered 2 days ago
ItsJustAMeasureBroItsJustAMeasureBro
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362
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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 f: E × E →V such that:
(a) f(x, y) = f(x, z) + f(z, y), x, y, z∈E and
(b) For every x∈E and v∈V there exists a unique element y∈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.
answered 2 days ago
user21793user21793
1492
add a comment |
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 f: E × E →V such that:
(a) f(x, y) = f(x, z) + f(z, y), x, y, z∈E and
(b) For every x∈E and v∈V there exists a unique element y∈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.
answered 2 days ago
user21793user21793
1492
add a comment |
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 f: E × E →V such that:
(a) f(x, y) = f(x, z) + f(z, y), x, y, z∈E and
(b) For every x∈E and v∈V there exists a unique element y∈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.
answered 2 days ago
user21793user21793
1492
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 f: E × E →V such that:
(a) f(x, y) = f(x, z) + f(z, y), x, y, z∈E and
(b) For every x∈E and v∈V there exists a unique element y∈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.
answered 2 days ago
user21793user21793
1492
answered 2 days ago
user21793user21793
1492
answered 2 days ago
user21793user21793
1492
answered 2 days ago
user21793user21793
1492
1492
add a comment |
add a comment |
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1
The product topology and metric topology are the same, in this case.
– Cameron Buie
2 days ago
All topologies generated by a norm on a finite-dimensional space over the real or complex numbers are in fact the same.
– ncmathsadist
2 days ago
So can I use the definition I have provided? (Sorry for the bluntness)
– Aryaman Gupta
2 days ago
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!
– Calum Gilhooley
2 days ago
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? .........
– Lee Mosher
2 days ago