Convergence of a Sequence - metric explanation
I am new to functional Analysis and the way convergence of a sequence is defined confuses me . I am reading the book by Kreyzig where he says :
A sequence ${(x_n)}$ in a metric space $X =(X,d)$ is said to converge if there is an $x in X$ such that
$$ lim_{n to inf} d(x_{n},x) = 0$$
x is called the limit of $x_{n}$ and we write :
$$ lim_{n to inf} x_{n} = x$$
Further , the author says that the metric d yields the sequence of real numbers : $$a_{n} = d(x_{n},x)$$ whose convergence defines that of $x_{n}$.
My Questions are
1.) When author says $x in X$ , does it mean $x$ is a sequence ? Intuitively $x$ should be a number as it is the limit of a sequence . BUT
2.) If $x$ is a number then how can we calculate the distance which is defined on sequences ,given that we are in a sequence space.
3.)How can the metric $d$ yield a sequence $a_{n}$ and not a number ?
real-analysis functional-analysis analysis convergence
asked Nov 29 at 19:03
Abhishek_04
264
add a comment |
I am new to functional Analysis and the way convergence of a sequence is defined confuses me . I am reading the book by Kreyzig where he says :
A sequence ${(x_n)}$ in a metric space $X =(X,d)$ is said to converge if there is an $x in X$ such that
$$ lim_{n to inf} d(x_{n},x) = 0$$
x is called the limit of $x_{n}$ and we write :
$$ lim_{n to inf} x_{n} = x$$
Further , the author says that the metric d yields the sequence of real numbers : $$a_{n} = d(x_{n},x)$$ whose convergence defines that of $x_{n}$.
My Questions are
1.) When author says $x in X$ , does it mean $x$ is a sequence ? Intuitively $x$ should be a number as it is the limit of a sequence . BUT
2.) If $x$ is a number then how can we calculate the distance which is defined on sequences ,given that we are in a sequence space.
3.)How can the metric $d$ yield a sequence $a_{n}$ and not a number ?
real-analysis functional-analysis analysis convergence
asked Nov 29 at 19:03
Abhishek_04
264
add a comment |
I am new to functional Analysis and the way convergence of a sequence is defined confuses me . I am reading the book by Kreyzig where he says :
A sequence ${(x_n)}$ in a metric space $X =(X,d)$ is said to converge if there is an $x in X$ such that
$$ lim_{n to inf} d(x_{n},x) = 0$$
x is called the limit of $x_{n}$ and we write :
$$ lim_{n to inf} x_{n} = x$$
Further , the author says that the metric d yields the sequence of real numbers : $$a_{n} = d(x_{n},x)$$ whose convergence defines that of $x_{n}$.
My Questions are
1.) When author says $x in X$ , does it mean $x$ is a sequence ? Intuitively $x$ should be a number as it is the limit of a sequence . BUT
2.) If $x$ is a number then how can we calculate the distance which is defined on sequences ,given that we are in a sequence space.
3.)How can the metric $d$ yield a sequence $a_{n}$ and not a number ?
real-analysis functional-analysis analysis convergence
asked Nov 29 at 19:03
Abhishek_04
264
I am new to functional Analysis and the way convergence of a sequence is defined confuses me . I am reading the book by Kreyzig where he says :
A sequence ${(x_n)}$ in a metric space $X =(X,d)$ is said to converge if there is an $x in X$ such that
$$ lim_{n to inf} d(x_{n},x) = 0$$
x is called the limit of $x_{n}$ and we write :
$$ lim_{n to inf} x_{n} = x$$
Further , the author says that the metric d yields the sequence of real numbers : $$a_{n} = d(x_{n},x)$$ whose convergence defines that of $x_{n}$.
My Questions are
1.) When author says $x in X$ , does it mean $x$ is a sequence ? Intuitively $x$ should be a number as it is the limit of a sequence . BUT
2.) If $x$ is a number then how can we calculate the distance which is defined on sequences ,given that we are in a sequence space.
3.)How can the metric $d$ yield a sequence $a_{n}$ and not a number ?
real-analysis functional-analysis analysis convergence
real-analysis functional-analysis analysis convergence
asked Nov 29 at 19:03
Abhishek_04
264
asked Nov 29 at 19:03
Abhishek_04
264
edited Nov 29 at 19:13
asked Nov 29 at 19:03
Abhishek_04
264
asked Nov 29 at 19:03
Abhishek_04
264
asked Nov 29 at 19:03
Abhishek_04
264
264
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
1.) x is an element of X. If X is $mathbb{R}$ then yes, x is just one real number.
To see why we care about $x in mathbb{R}$, consider if $X = (0,1)$. A sequence like $frac{1}{n}$ will not (and should intuitively) not converge because $0$ is not in the set $X$.
2/3.) Kreyzig is just defining a new sequence ${a_n}$. Every metric space will have some metric d. It might be helpful to just think about this as 'distance'. How he is defining ${a_n}$ is by taking the distance from every point in the sequence ${x_n}$ and subtracting the point to which the sequence converges. See that this sequence should then converge to $0$.
Note when he writes $a_n = d(x_n, x)$ he is defining a term of the sequence. Hence ${a_n} = {a_1, a_2,... }$.
answered Nov 29 at 19:22
T. Ford
12118
What If X is a general sequence space not specifially R ? what about x then ?
– Abhishek_04
Nov 29 at 19:31
I understand that metric defines the notion of a distance . If we , for example, talk of an L-P space, then we have have a well defined metric which gives a number. But if want to talk of convergence of an element of that sequence space, we define another distance metric given in the book . is that correct ?
– Abhishek_04
Nov 29 at 19:35
So I'm not sure what your book says, but it might use the p-norm. I have also seen the metric $lim |m_n - k_n|$ for the space of Cauchy sequences over $mathbb{Q}$. But in general, yes it definitely depends on the metric $d$ and some might define it different than others.
– T. Ford
Nov 29 at 19:47
add a comment |
SO i finally figured out what mistake i was doing .
1.) A sequence can be defined in any metric space (X,d) where X does not necessarily have to be space of sequence like the $l^{p}$ . So for the case when X is R , x wil just be a number, if X is a sequence space , the limit will indeed be a sequence.
answered Nov 30 at 7:42
Abhishek_04
264
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
1.) x is an element of X. If X is $mathbb{R}$ then yes, x is just one real number.
To see why we care about $x in mathbb{R}$, consider if $X = (0,1)$. A sequence like $frac{1}{n}$ will not (and should intuitively) not converge because $0$ is not in the set $X$.
2/3.) Kreyzig is just defining a new sequence ${a_n}$. Every metric space will have some metric d. It might be helpful to just think about this as 'distance'. How he is defining ${a_n}$ is by taking the distance from every point in the sequence ${x_n}$ and subtracting the point to which the sequence converges. See that this sequence should then converge to $0$.
Note when he writes $a_n = d(x_n, x)$ he is defining a term of the sequence. Hence ${a_n} = {a_1, a_2,... }$.
answered Nov 29 at 19:22
T. Ford
12118
What If X is a general sequence space not specifially R ? what about x then ?
– Abhishek_04
Nov 29 at 19:31
I understand that metric defines the notion of a distance . If we , for example, talk of an L-P space, then we have have a well defined metric which gives a number. But if want to talk of convergence of an element of that sequence space, we define another distance metric given in the book . is that correct ?
– Abhishek_04
Nov 29 at 19:35
So I'm not sure what your book says, but it might use the p-norm. I have also seen the metric $lim |m_n - k_n|$ for the space of Cauchy sequences over $mathbb{Q}$. But in general, yes it definitely depends on the metric $d$ and some might define it different than others.
– T. Ford
Nov 29 at 19:47
add a comment |
1.) x is an element of X. If X is $mathbb{R}$ then yes, x is just one real number.
To see why we care about $x in mathbb{R}$, consider if $X = (0,1)$. A sequence like $frac{1}{n}$ will not (and should intuitively) not converge because $0$ is not in the set $X$.
2/3.) Kreyzig is just defining a new sequence ${a_n}$. Every metric space will have some metric d. It might be helpful to just think about this as 'distance'. How he is defining ${a_n}$ is by taking the distance from every point in the sequence ${x_n}$ and subtracting the point to which the sequence converges. See that this sequence should then converge to $0$.
Note when he writes $a_n = d(x_n, x)$ he is defining a term of the sequence. Hence ${a_n} = {a_1, a_2,... }$.
answered Nov 29 at 19:22
T. Ford
12118
What If X is a general sequence space not specifially R ? what about x then ?
– Abhishek_04
Nov 29 at 19:31
I understand that metric defines the notion of a distance . If we , for example, talk of an L-P space, then we have have a well defined metric which gives a number. But if want to talk of convergence of an element of that sequence space, we define another distance metric given in the book . is that correct ?
– Abhishek_04
Nov 29 at 19:35
So I'm not sure what your book says, but it might use the p-norm. I have also seen the metric $lim |m_n - k_n|$ for the space of Cauchy sequences over $mathbb{Q}$. But in general, yes it definitely depends on the metric $d$ and some might define it different than others.
– T. Ford
Nov 29 at 19:47
add a comment |
1.) x is an element of X. If X is $mathbb{R}$ then yes, x is just one real number.
To see why we care about $x in mathbb{R}$, consider if $X = (0,1)$. A sequence like $frac{1}{n}$ will not (and should intuitively) not converge because $0$ is not in the set $X$.
2/3.) Kreyzig is just defining a new sequence ${a_n}$. Every metric space will have some metric d. It might be helpful to just think about this as 'distance'. How he is defining ${a_n}$ is by taking the distance from every point in the sequence ${x_n}$ and subtracting the point to which the sequence converges. See that this sequence should then converge to $0$.
Note when he writes $a_n = d(x_n, x)$ he is defining a term of the sequence. Hence ${a_n} = {a_1, a_2,... }$.
answered Nov 29 at 19:22
T. Ford
12118
1.) x is an element of X. If X is $mathbb{R}$ then yes, x is just one real number.
To see why we care about $x in mathbb{R}$, consider if $X = (0,1)$. A sequence like $frac{1}{n}$ will not (and should intuitively) not converge because $0$ is not in the set $X$.
2/3.) Kreyzig is just defining a new sequence ${a_n}$. Every metric space will have some metric d. It might be helpful to just think about this as 'distance'. How he is defining ${a_n}$ is by taking the distance from every point in the sequence ${x_n}$ and subtracting the point to which the sequence converges. See that this sequence should then converge to $0$.
Note when he writes $a_n = d(x_n, x)$ he is defining a term of the sequence. Hence ${a_n} = {a_1, a_2,... }$.
answered Nov 29 at 19:22
T. Ford
12118
answered Nov 29 at 19:22
T. Ford
12118
answered Nov 29 at 19:22
T. Ford
12118
answered Nov 29 at 19:22
T. Ford
12118
12118
What If X is a general sequence space not specifially R ? what about x then ?
– Abhishek_04
Nov 29 at 19:31
I understand that metric defines the notion of a distance . If we , for example, talk of an L-P space, then we have have a well defined metric which gives a number. But if want to talk of convergence of an element of that sequence space, we define another distance metric given in the book . is that correct ?
– Abhishek_04
Nov 29 at 19:35
So I'm not sure what your book says, but it might use the p-norm. I have also seen the metric $lim |m_n - k_n|$ for the space of Cauchy sequences over $mathbb{Q}$. But in general, yes it definitely depends on the metric $d$ and some might define it different than others.
– T. Ford
Nov 29 at 19:47
add a comment |
What If X is a general sequence space not specifially R ? what about x then ?
– Abhishek_04
Nov 29 at 19:31
I understand that metric defines the notion of a distance . If we , for example, talk of an L-P space, then we have have a well defined metric which gives a number. But if want to talk of convergence of an element of that sequence space, we define another distance metric given in the book . is that correct ?
– Abhishek_04
Nov 29 at 19:35
So I'm not sure what your book says, but it might use the p-norm. I have also seen the metric $lim |m_n - k_n|$ for the space of Cauchy sequences over $mathbb{Q}$. But in general, yes it definitely depends on the metric $d$ and some might define it different than others.
– T. Ford
Nov 29 at 19:47
What If X is a general sequence space not specifially R ? what about x then ?
– Abhishek_04
Nov 29 at 19:31
What If X is a general sequence space not specifially R ? what about x then ?
– Abhishek_04
Nov 29 at 19:31
I understand that metric defines the notion of a distance . If we , for example, talk of an L-P space, then we have have a well defined metric which gives a number. But if want to talk of convergence of an element of that sequence space, we define another distance metric given in the book . is that correct ?
– Abhishek_04
Nov 29 at 19:35
I understand that metric defines the notion of a distance . If we , for example, talk of an L-P space, then we have have a well defined metric which gives a number. But if want to talk of convergence of an element of that sequence space, we define another distance metric given in the book . is that correct ?
– Abhishek_04
Nov 29 at 19:35
So I'm not sure what your book says, but it might use the p-norm. I have also seen the metric $lim |m_n - k_n|$ for the space of Cauchy sequences over $mathbb{Q}$. But in general, yes it definitely depends on the metric $d$ and some might define it different than others.
– T. Ford
Nov 29 at 19:47
So I'm not sure what your book says, but it might use the p-norm. I have also seen the metric $lim |m_n - k_n|$ for the space of Cauchy sequences over $mathbb{Q}$. But in general, yes it definitely depends on the metric $d$ and some might define it different than others.
– T. Ford
Nov 29 at 19:47
add a comment |
SO i finally figured out what mistake i was doing .
1.) A sequence can be defined in any metric space (X,d) where X does not necessarily have to be space of sequence like the $l^{p}$ . So for the case when X is R , x wil just be a number, if X is a sequence space , the limit will indeed be a sequence.
answered Nov 30 at 7:42
Abhishek_04
264
add a comment |
SO i finally figured out what mistake i was doing .
1.) A sequence can be defined in any metric space (X,d) where X does not necessarily have to be space of sequence like the $l^{p}$ . So for the case when X is R , x wil just be a number, if X is a sequence space , the limit will indeed be a sequence.
answered Nov 30 at 7:42
Abhishek_04
264
add a comment |
SO i finally figured out what mistake i was doing .
1.) A sequence can be defined in any metric space (X,d) where X does not necessarily have to be space of sequence like the $l^{p}$ . So for the case when X is R , x wil just be a number, if X is a sequence space , the limit will indeed be a sequence.
answered Nov 30 at 7:42
Abhishek_04
264
SO i finally figured out what mistake i was doing .
1.) A sequence can be defined in any metric space (X,d) where X does not necessarily have to be space of sequence like the $l^{p}$ . So for the case when X is R , x wil just be a number, if X is a sequence space , the limit will indeed be a sequence.
answered Nov 30 at 7:42
Abhishek_04
264
answered Nov 30 at 7:42
Abhishek_04
264
answered Nov 30 at 7:42
Abhishek_04
264
answered Nov 30 at 7:42
Abhishek_04
264
264
add a comment |
add a comment |
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