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What does this vector space mean? [closed]
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Here is a quote from pages 4 and 5 of https://aetos.it.teithe.gr/~gouliana/en/2017-polynomials.pdf
As it is well known from nonlinear algebra, the structure of a typical nonlinear algebraic system of $n$ equations with $n$ unknowns has the form
$$f_1(x_1, x_2, ..., x_n)=0$$
$$f_2(x_1, x_2, ..., x_n)=0$$
$$cdots$$
$$f_n(x_1, x_2, ..., x_n)=0$$
Or using vector notation, $F(x)=0$ where
$$F=(f_1, f_2, ..., f_n)^T$$
is a vector of non-linear functions $f_i(x)=f_i(x_1, x_2, ..., x_n)$
each being defined in the vector space
$$Omega=prod_i=1^nalpha_i, beta_i subset textR^n$$
of all real valued and continuous functions.
My question is what does the above vector space mean?
abstract-algebra functions vector-spaces
asked Jul 26 at 20:27
Shrey Joshi
1389
closed as unclear what you're asking by Peter Franek, M. Winter, Anubhav Mukherjee, amWhy, Adrian Keister Jul 27 at 0:17
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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up vote
1
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Here is a quote from pages 4 and 5 of https://aetos.it.teithe.gr/~gouliana/en/2017-polynomials.pdf
As it is well known from nonlinear algebra, the structure of a typical nonlinear algebraic system of $n$ equations with $n$ unknowns has the form
$$f_1(x_1, x_2, ..., x_n)=0$$
$$f_2(x_1, x_2, ..., x_n)=0$$
$$cdots$$
$$f_n(x_1, x_2, ..., x_n)=0$$
Or using vector notation, $F(x)=0$ where
$$F=(f_1, f_2, ..., f_n)^T$$
is a vector of non-linear functions $f_i(x)=f_i(x_1, x_2, ..., x_n)$
each being defined in the vector space
$$Omega=prod_i=1^nalpha_i, beta_i subset textR^n$$
of all real valued and continuous functions.
My question is what does the above vector space mean?
abstract-algebra functions vector-spaces
asked Jul 26 at 20:27
Shrey Joshi
1389
closed as unclear what you're asking by Peter Franek, M. Winter, Anubhav Mukherjee, amWhy, Adrian Keister Jul 27 at 0:17
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
I assume that $alpha_i, beta_i$ is supposed to represent an interval. Aren't there some kind of brackets around it? But in any event, this isn't a vector space with the usual definition of addition. If the source says it's a vector space, it must define what addition and scalar multiplication mean.
– saulspatz
Jul 26 at 20:32
2
What you've written makes no sense to me. I can't tell what things are in $Omega$. Please edit the question to include more context for the source of the problem.
– Ethan Bolker
Jul 26 at 20:32
As written, it is not clear what that means
– Peter Franek
Jul 26 at 20:32
2
The source says $alpha_i, beta_i$
– Shrey Joshi
Jul 26 at 20:46
2
Seems like a mistake. Overall, the text does not impress me as a hight-quality text, maybe you should find a better textbook.
– Peter Franek
Jul 26 at 20:53
 |Â
show 6 more comments
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Here is a quote from pages 4 and 5 of https://aetos.it.teithe.gr/~gouliana/en/2017-polynomials.pdf
As it is well known from nonlinear algebra, the structure of a typical nonlinear algebraic system of $n$ equations with $n$ unknowns has the form
$$f_1(x_1, x_2, ..., x_n)=0$$
$$f_2(x_1, x_2, ..., x_n)=0$$
$$cdots$$
$$f_n(x_1, x_2, ..., x_n)=0$$
Or using vector notation, $F(x)=0$ where
$$F=(f_1, f_2, ..., f_n)^T$$
is a vector of non-linear functions $f_i(x)=f_i(x_1, x_2, ..., x_n)$
each being defined in the vector space
$$Omega=prod_i=1^nalpha_i, beta_i subset textR^n$$
of all real valued and continuous functions.
My question is what does the above vector space mean?
abstract-algebra functions vector-spaces
asked Jul 26 at 20:27
Shrey Joshi
1389
Here is a quote from pages 4 and 5 of https://aetos.it.teithe.gr/~gouliana/en/2017-polynomials.pdf
As it is well known from nonlinear algebra, the structure of a typical nonlinear algebraic system of $n$ equations with $n$ unknowns has the form
$$f_1(x_1, x_2, ..., x_n)=0$$
$$f_2(x_1, x_2, ..., x_n)=0$$
$$cdots$$
$$f_n(x_1, x_2, ..., x_n)=0$$
Or using vector notation, $F(x)=0$ where
$$F=(f_1, f_2, ..., f_n)^T$$
is a vector of non-linear functions $f_i(x)=f_i(x_1, x_2, ..., x_n)$
each being defined in the vector space
$$Omega=prod_i=1^nalpha_i, beta_i subset textR^n$$
of all real valued and continuous functions.
My question is what does the above vector space mean?
abstract-algebra functions vector-spaces
asked Jul 26 at 20:27
Shrey Joshi
1389
edited Jul 26 at 20:51
asked Jul 26 at 20:27
Shrey Joshi
1389
asked Jul 26 at 20:27
Shrey Joshi
1389
asked Jul 26 at 20:27
Shrey Joshi
1389
1389
closed as unclear what you're asking by Peter Franek, M. Winter, Anubhav Mukherjee, amWhy, Adrian Keister Jul 27 at 0:17
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Peter Franek, M. Winter, Anubhav Mukherjee, amWhy, Adrian Keister Jul 27 at 0:17
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
I assume that $alpha_i, beta_i$ is supposed to represent an interval. Aren't there some kind of brackets around it? But in any event, this isn't a vector space with the usual definition of addition. If the source says it's a vector space, it must define what addition and scalar multiplication mean.
– saulspatz
Jul 26 at 20:32
2
What you've written makes no sense to me. I can't tell what things are in $Omega$. Please edit the question to include more context for the source of the problem.
– Ethan Bolker
Jul 26 at 20:32
As written, it is not clear what that means
– Peter Franek
Jul 26 at 20:32
2
The source says $alpha_i, beta_i$
– Shrey Joshi
Jul 26 at 20:46
2
Seems like a mistake. Overall, the text does not impress me as a hight-quality text, maybe you should find a better textbook.
– Peter Franek
Jul 26 at 20:53
 |Â
show 6 more comments
I assume that $alpha_i, beta_i$ is supposed to represent an interval. Aren't there some kind of brackets around it? But in any event, this isn't a vector space with the usual definition of addition. If the source says it's a vector space, it must define what addition and scalar multiplication mean.
– saulspatz
Jul 26 at 20:32
2
What you've written makes no sense to me. I can't tell what things are in $Omega$. Please edit the question to include more context for the source of the problem.
– Ethan Bolker
Jul 26 at 20:32
As written, it is not clear what that means
– Peter Franek
Jul 26 at 20:32
2
The source says $alpha_i, beta_i$
– Shrey Joshi
Jul 26 at 20:46
2
Seems like a mistake. Overall, the text does not impress me as a hight-quality text, maybe you should find a better textbook.
– Peter Franek
Jul 26 at 20:53
I assume that $alpha_i, beta_i$ is supposed to represent an interval. Aren't there some kind of brackets around it? But in any event, this isn't a vector space with the usual definition of addition. If the source says it's a vector space, it must define what addition and scalar multiplication mean.
– saulspatz
Jul 26 at 20:32
I assume that $alpha_i, beta_i$ is supposed to represent an interval. Aren't there some kind of brackets around it? But in any event, this isn't a vector space with the usual definition of addition. If the source says it's a vector space, it must define what addition and scalar multiplication mean.
– saulspatz
Jul 26 at 20:32
2
2
What you've written makes no sense to me. I can't tell what things are in $Omega$. Please edit the question to include more context for the source of the problem.
– Ethan Bolker
Jul 26 at 20:32
What you've written makes no sense to me. I can't tell what things are in $Omega$. Please edit the question to include more context for the source of the problem.
– Ethan Bolker
Jul 26 at 20:32
As written, it is not clear what that means
– Peter Franek
Jul 26 at 20:32
As written, it is not clear what that means
– Peter Franek
Jul 26 at 20:32
2
2
The source says $alpha_i, beta_i$
– Shrey Joshi
Jul 26 at 20:46
The source says $alpha_i, beta_i$
– Shrey Joshi
Jul 26 at 20:46
2
2
Seems like a mistake. Overall, the text does not impress me as a hight-quality text, maybe you should find a better textbook.
– Peter Franek
Jul 26 at 20:53
Seems like a mistake. Overall, the text does not impress me as a hight-quality text, maybe you should find a better textbook.
– Peter Franek
Jul 26 at 20:53
 |Â
show 6 more comments
1 Answer
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From the context, he means that each of the coordinate functions $f_i$ is defined and continuous on $$Omega=prod_i=1^n[alpha_i, beta_i] subset textR^n$$
The vector space referred to is the vector of space of such functions.
To put it another way, $$
Omega=alpha_ileq x_ileqbeta_i, i=1,2,dots,n
$$
This isn't what the paper says, but I feel pretty sure that's what it means. I'm not certain, though, about the inequalities. Perhaps they are meant to be strict.
answered Jul 26 at 21:08
saulspatz
10.4k21323
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
From the context, he means that each of the coordinate functions $f_i$ is defined and continuous on $$Omega=prod_i=1^n[alpha_i, beta_i] subset textR^n$$
The vector space referred to is the vector of space of such functions.
To put it another way, $$
Omega=alpha_ileq x_ileqbeta_i, i=1,2,dots,n
$$
This isn't what the paper says, but I feel pretty sure that's what it means. I'm not certain, though, about the inequalities. Perhaps they are meant to be strict.
answered Jul 26 at 21:08
saulspatz
10.4k21323
add a comment |Â
up vote
0
down vote
accepted
From the context, he means that each of the coordinate functions $f_i$ is defined and continuous on $$Omega=prod_i=1^n[alpha_i, beta_i] subset textR^n$$
The vector space referred to is the vector of space of such functions.
To put it another way, $$
Omega=alpha_ileq x_ileqbeta_i, i=1,2,dots,n
$$
This isn't what the paper says, but I feel pretty sure that's what it means. I'm not certain, though, about the inequalities. Perhaps they are meant to be strict.
answered Jul 26 at 21:08
saulspatz
10.4k21323
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
From the context, he means that each of the coordinate functions $f_i$ is defined and continuous on $$Omega=prod_i=1^n[alpha_i, beta_i] subset textR^n$$
The vector space referred to is the vector of space of such functions.
To put it another way, $$
Omega=alpha_ileq x_ileqbeta_i, i=1,2,dots,n
$$
This isn't what the paper says, but I feel pretty sure that's what it means. I'm not certain, though, about the inequalities. Perhaps they are meant to be strict.
answered Jul 26 at 21:08
saulspatz
10.4k21323
From the context, he means that each of the coordinate functions $f_i$ is defined and continuous on $$Omega=prod_i=1^n[alpha_i, beta_i] subset textR^n$$
The vector space referred to is the vector of space of such functions.
To put it another way, $$
Omega=alpha_ileq x_ileqbeta_i, i=1,2,dots,n
$$
This isn't what the paper says, but I feel pretty sure that's what it means. I'm not certain, though, about the inequalities. Perhaps they are meant to be strict.
answered Jul 26 at 21:08
saulspatz
10.4k21323
answered Jul 26 at 21:08
saulspatz
10.4k21323
answered Jul 26 at 21:08
saulspatz
10.4k21323
answered Jul 26 at 21:08
saulspatz
10.4k21323
10.4k21323
add a comment |Â
add a comment |Â
I assume that $alpha_i, beta_i$ is supposed to represent an interval. Aren't there some kind of brackets around it? But in any event, this isn't a vector space with the usual definition of addition. If the source says it's a vector space, it must define what addition and scalar multiplication mean.
– saulspatz
Jul 26 at 20:32
2
What you've written makes no sense to me. I can't tell what things are in $Omega$. Please edit the question to include more context for the source of the problem.
– Ethan Bolker
Jul 26 at 20:32
As written, it is not clear what that means
– Peter Franek
Jul 26 at 20:32
2
The source says $alpha_i, beta_i$
– Shrey Joshi
Jul 26 at 20:46
2
Seems like a mistake. Overall, the text does not impress me as a hight-quality text, maybe you should find a better textbook.
– Peter Franek
Jul 26 at 20:53