Is this seminormed space (intersection of Banach spaces) non-empty?
$begingroup$
In this paper of Zubelevich the author consider a a scale of Banach spaces ${(E_{s},|cdot|_{s}):0<s<1}$, that is to say, each $(E_{s},|cdot|_{s})$ is a Banach space and
$$
E_{s+delta}subseteq E_{s},quad |cdot|_{s}leq |cdot|_{s+delta}, quad s+delta<1, delta>0,
$$
for each $0<s<1$. Then, fixed $a>1$, defines the (open) triangle
$$
Delta :={(tau,s)inmathbb{R}^{2}:tau>0,0<s<1,1-s-atau>0 },
$$
and the seminormed space
$$
E:=bigcap_{(s,tau)in Delta} C([0,tau],E_{s}),
$$
endowed the family of norms $|u|_{tau,s}:=max_{0leq tleq tau}|u(t)|_{s}$. As usual, $C([0,tau],E_{s})$ denotes the space of the continuous maps $u:[0,tau]longrightarrow (E_{s},|cdot|_{s})$. So, $E$ is a (locally convex) topological space with a basis of the topology given by the open balls $B_{tau,s}:={uin E: |u|_{tau,s}<r}$, for each $r>0$.
My question is the following: Is $E$ non-empty? Or, more precisely, under that conditions $E$ is not a "trivial" (for instance, a single point) space?
Thank you very much in advance for your comments.
general-topology functional-analysis analysis
asked Jan 10 at 7:08
user123043user123043
126110
$endgroup$
add a comment |
$begingroup$
In this paper of Zubelevich the author consider a a scale of Banach spaces ${(E_{s},|cdot|_{s}):0<s<1}$, that is to say, each $(E_{s},|cdot|_{s})$ is a Banach space and
$$
E_{s+delta}subseteq E_{s},quad |cdot|_{s}leq |cdot|_{s+delta}, quad s+delta<1, delta>0,
$$
for each $0<s<1$. Then, fixed $a>1$, defines the (open) triangle
$$
Delta :={(tau,s)inmathbb{R}^{2}:tau>0,0<s<1,1-s-atau>0 },
$$
and the seminormed space
$$
E:=bigcap_{(s,tau)in Delta} C([0,tau],E_{s}),
$$
endowed the family of norms $|u|_{tau,s}:=max_{0leq tleq tau}|u(t)|_{s}$. As usual, $C([0,tau],E_{s})$ denotes the space of the continuous maps $u:[0,tau]longrightarrow (E_{s},|cdot|_{s})$. So, $E$ is a (locally convex) topological space with a basis of the topology given by the open balls $B_{tau,s}:={uin E: |u|_{tau,s}<r}$, for each $r>0$.
My question is the following: Is $E$ non-empty? Or, more precisely, under that conditions $E$ is not a "trivial" (for instance, a single point) space?
Thank you very much in advance for your comments.
general-topology functional-analysis analysis
asked Jan 10 at 7:08
user123043user123043
126110
$endgroup$
$begingroup$
Aren't constant functions that have values in $E_1$ contained in $E$? Also, what is the intersection of $C([0,tau_1],E_{s_1}$ with $C([0,tau_2],E_{s_2}$ for $tau_1netau_2$?
$endgroup$
– daw
Jan 10 at 7:32
$begingroup$
Thanks @daw. $E_{1}$ "a priori" could not be a Banach space, not? i.e, may not be "well defined" in the escale. I do not undestand at all your second question...
$endgroup$
– user123043
Jan 10 at 8:18
1
$begingroup$
How do you define the intersection of function spaces, where the functions have different domain? I.e, what is $C([0,1];mathbb R) cap C([0,2],mathbb R)$.
$endgroup$
– daw
Jan 10 at 11:15
$begingroup$
Yes, I agree with you, How it can be defined that intersection? I am reading the cited paper, but I feel that the definition of E can has "some hole", no? Thanks @daw
$endgroup$
– user123043
Jan 10 at 11:28
$begingroup$
It seems I met the author at Russian scientific forum where he is very active and writes different things. So I think there are big chances that he will answer your letter.
$endgroup$
– Alex Ravsky
Jan 10 at 15:45
add a comment |
$begingroup$
In this paper of Zubelevich the author consider a a scale of Banach spaces ${(E_{s},|cdot|_{s}):0<s<1}$, that is to say, each $(E_{s},|cdot|_{s})$ is a Banach space and
$$
E_{s+delta}subseteq E_{s},quad |cdot|_{s}leq |cdot|_{s+delta}, quad s+delta<1, delta>0,
$$
for each $0<s<1$. Then, fixed $a>1$, defines the (open) triangle
$$
Delta :={(tau,s)inmathbb{R}^{2}:tau>0,0<s<1,1-s-atau>0 },
$$
and the seminormed space
$$
E:=bigcap_{(s,tau)in Delta} C([0,tau],E_{s}),
$$
endowed the family of norms $|u|_{tau,s}:=max_{0leq tleq tau}|u(t)|_{s}$. As usual, $C([0,tau],E_{s})$ denotes the space of the continuous maps $u:[0,tau]longrightarrow (E_{s},|cdot|_{s})$. So, $E$ is a (locally convex) topological space with a basis of the topology given by the open balls $B_{tau,s}:={uin E: |u|_{tau,s}<r}$, for each $r>0$.
My question is the following: Is $E$ non-empty? Or, more precisely, under that conditions $E$ is not a "trivial" (for instance, a single point) space?
Thank you very much in advance for your comments.
general-topology functional-analysis analysis
asked Jan 10 at 7:08
user123043user123043
126110
$endgroup$
In this paper of Zubelevich the author consider a a scale of Banach spaces ${(E_{s},|cdot|_{s}):0<s<1}$, that is to say, each $(E_{s},|cdot|_{s})$ is a Banach space and
$$
E_{s+delta}subseteq E_{s},quad |cdot|_{s}leq |cdot|_{s+delta}, quad s+delta<1, delta>0,
$$
for each $0<s<1$. Then, fixed $a>1$, defines the (open) triangle
$$
Delta :={(tau,s)inmathbb{R}^{2}:tau>0,0<s<1,1-s-atau>0 },
$$
and the seminormed space
$$
E:=bigcap_{(s,tau)in Delta} C([0,tau],E_{s}),
$$
endowed the family of norms $|u|_{tau,s}:=max_{0leq tleq tau}|u(t)|_{s}$. As usual, $C([0,tau],E_{s})$ denotes the space of the continuous maps $u:[0,tau]longrightarrow (E_{s},|cdot|_{s})$. So, $E$ is a (locally convex) topological space with a basis of the topology given by the open balls $B_{tau,s}:={uin E: |u|_{tau,s}<r}$, for each $r>0$.
My question is the following: Is $E$ non-empty? Or, more precisely, under that conditions $E$ is not a "trivial" (for instance, a single point) space?
Thank you very much in advance for your comments.
general-topology functional-analysis analysis
general-topology functional-analysis analysis
asked Jan 10 at 7:08
user123043user123043
126110
asked Jan 10 at 7:08
user123043user123043
126110
asked Jan 10 at 7:08
user123043user123043
126110
asked Jan 10 at 7:08
user123043user123043
126110
asked Jan 10 at 7:08
user123043user123043
126110
126110
$begingroup$
Aren't constant functions that have values in $E_1$ contained in $E$? Also, what is the intersection of $C([0,tau_1],E_{s_1}$ with $C([0,tau_2],E_{s_2}$ for $tau_1netau_2$?
$endgroup$
– daw
Jan 10 at 7:32
$begingroup$
Thanks @daw. $E_{1}$ "a priori" could not be a Banach space, not? i.e, may not be "well defined" in the escale. I do not undestand at all your second question...
$endgroup$
– user123043
Jan 10 at 8:18
1
$begingroup$
How do you define the intersection of function spaces, where the functions have different domain? I.e, what is $C([0,1];mathbb R) cap C([0,2],mathbb R)$.
$endgroup$
– daw
Jan 10 at 11:15
$begingroup$
Yes, I agree with you, How it can be defined that intersection? I am reading the cited paper, but I feel that the definition of E can has "some hole", no? Thanks @daw
$endgroup$
– user123043
Jan 10 at 11:28
$begingroup$
It seems I met the author at Russian scientific forum where he is very active and writes different things. So I think there are big chances that he will answer your letter.
$endgroup$
– Alex Ravsky
Jan 10 at 15:45
add a comment |
$begingroup$
Aren't constant functions that have values in $E_1$ contained in $E$? Also, what is the intersection of $C([0,tau_1],E_{s_1}$ with $C([0,tau_2],E_{s_2}$ for $tau_1netau_2$?
$endgroup$
– daw
Jan 10 at 7:32
$begingroup$
Thanks @daw. $E_{1}$ "a priori" could not be a Banach space, not? i.e, may not be "well defined" in the escale. I do not undestand at all your second question...
$endgroup$
– user123043
Jan 10 at 8:18
1
$begingroup$
How do you define the intersection of function spaces, where the functions have different domain? I.e, what is $C([0,1];mathbb R) cap C([0,2],mathbb R)$.
$endgroup$
– daw
Jan 10 at 11:15
$begingroup$
Yes, I agree with you, How it can be defined that intersection? I am reading the cited paper, but I feel that the definition of E can has "some hole", no? Thanks @daw
$endgroup$
– user123043
Jan 10 at 11:28
$begingroup$
It seems I met the author at Russian scientific forum where he is very active and writes different things. So I think there are big chances that he will answer your letter.
$endgroup$
– Alex Ravsky
Jan 10 at 15:45
$begingroup$
Aren't constant functions that have values in $E_1$ contained in $E$? Also, what is the intersection of $C([0,tau_1],E_{s_1}$ with $C([0,tau_2],E_{s_2}$ for $tau_1netau_2$?
$endgroup$
– daw
Jan 10 at 7:32
$begingroup$
Aren't constant functions that have values in $E_1$ contained in $E$? Also, what is the intersection of $C([0,tau_1],E_{s_1}$ with $C([0,tau_2],E_{s_2}$ for $tau_1netau_2$?
$endgroup$
– daw
Jan 10 at 7:32
$begingroup$
Thanks @daw. $E_{1}$ "a priori" could not be a Banach space, not? i.e, may not be "well defined" in the escale. I do not undestand at all your second question...
$endgroup$
– user123043
Jan 10 at 8:18
$begingroup$
Thanks @daw. $E_{1}$ "a priori" could not be a Banach space, not? i.e, may not be "well defined" in the escale. I do not undestand at all your second question...
$endgroup$
– user123043
Jan 10 at 8:18
1
1
$begingroup$
How do you define the intersection of function spaces, where the functions have different domain? I.e, what is $C([0,1];mathbb R) cap C([0,2],mathbb R)$.
$endgroup$
– daw
Jan 10 at 11:15
$begingroup$
How do you define the intersection of function spaces, where the functions have different domain? I.e, what is $C([0,1];mathbb R) cap C([0,2],mathbb R)$.
$endgroup$
– daw
Jan 10 at 11:15
$begingroup$
Yes, I agree with you, How it can be defined that intersection? I am reading the cited paper, but I feel that the definition of E can has "some hole", no? Thanks @daw
$endgroup$
– user123043
Jan 10 at 11:28
$begingroup$
Yes, I agree with you, How it can be defined that intersection? I am reading the cited paper, but I feel that the definition of E can has "some hole", no? Thanks @daw
$endgroup$
– user123043
Jan 10 at 11:28
$begingroup$
It seems I met the author at Russian scientific forum where he is very active and writes different things. So I think there are big chances that he will answer your letter.
$endgroup$
– Alex Ravsky
Jan 10 at 15:45
$begingroup$
It seems I met the author at Russian scientific forum where he is very active and writes different things. So I think there are big chances that he will answer your letter.
$endgroup$
– Alex Ravsky
Jan 10 at 15:45
add a comment |
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$begingroup$
Aren't constant functions that have values in $E_1$ contained in $E$? Also, what is the intersection of $C([0,tau_1],E_{s_1}$ with $C([0,tau_2],E_{s_2}$ for $tau_1netau_2$?
$endgroup$
– daw
Jan 10 at 7:32
$begingroup$
Thanks @daw. $E_{1}$ "a priori" could not be a Banach space, not? i.e, may not be "well defined" in the escale. I do not undestand at all your second question...
$endgroup$
– user123043
Jan 10 at 8:18
1
$begingroup$
How do you define the intersection of function spaces, where the functions have different domain? I.e, what is $C([0,1];mathbb R) cap C([0,2],mathbb R)$.
$endgroup$
– daw
Jan 10 at 11:15
$begingroup$
Yes, I agree with you, How it can be defined that intersection? I am reading the cited paper, but I feel that the definition of E can has "some hole", no? Thanks @daw
$endgroup$
– user123043
Jan 10 at 11:28
$begingroup$
It seems I met the author at Russian scientific forum where he is very active and writes different things. So I think there are big chances that he will answer your letter.
$endgroup$
– Alex Ravsky
Jan 10 at 15:45