How to visualize the open ball in $C[0,1]$?
Clash Royale CLAN TAG#URR8PPP
up vote
3
down vote
favorite
Consider the metric space $C[0,1]$ with 'sup' metric:
$$d_infty(f,g)=textsup_x;vert f(x)-g(x)vert $$
How to visualize the open ball $B(f,r)$ where $f$ is the identity function and $r>0$?
Here, $B(f,r)=g in C[0,1]: textsup_x;vert x-g(x)vert<r. $
Is the figure look like the tube around the line $y=x$ ?
Added:
By DanielWainfleet answer, the figure look like this: (An outline with $r=1$)
Where the middle line is the identity function and and the upper line is the graph of $1+x$ and the bottom line is $-1+x$ .
A function $g in B(f,1)$ means $g$ lies entirely within this strip!
real-analysis metric-spaces
asked Aug 13 at 7:00
Learning Mathematics
543313
add a comment |Â
up vote
3
down vote
favorite
Consider the metric space $C[0,1]$ with 'sup' metric:
$$d_infty(f,g)=textsup_x;vert f(x)-g(x)vert $$
How to visualize the open ball $B(f,r)$ where $f$ is the identity function and $r>0$?
Here, $B(f,r)=g in C[0,1]: textsup_x;vert x-g(x)vert<r. $
Is the figure look like the tube around the line $y=x$ ?
Added:
By DanielWainfleet answer, the figure look like this: (An outline with $r=1$)
Where the middle line is the identity function and and the upper line is the graph of $1+x$ and the bottom line is $-1+x$ .
A function $g in B(f,1)$ means $g$ lies entirely within this strip!
real-analysis metric-spaces
asked Aug 13 at 7:00
Learning Mathematics
543313
1
Points in the ball are those functions whose graphs lie between the straight lines $y=x+r$ and $y =x-r$.
– Kavi Rama Murthy
Aug 13 at 7:20
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Consider the metric space $C[0,1]$ with 'sup' metric:
$$d_infty(f,g)=textsup_x;vert f(x)-g(x)vert $$
How to visualize the open ball $B(f,r)$ where $f$ is the identity function and $r>0$?
Here, $B(f,r)=g in C[0,1]: textsup_x;vert x-g(x)vert<r. $
Is the figure look like the tube around the line $y=x$ ?
Added:
By DanielWainfleet answer, the figure look like this: (An outline with $r=1$)
Where the middle line is the identity function and and the upper line is the graph of $1+x$ and the bottom line is $-1+x$ .
A function $g in B(f,1)$ means $g$ lies entirely within this strip!
real-analysis metric-spaces
asked Aug 13 at 7:00
Learning Mathematics
543313
Consider the metric space $C[0,1]$ with 'sup' metric:
$$d_infty(f,g)=textsup_x;vert f(x)-g(x)vert $$
How to visualize the open ball $B(f,r)$ where $f$ is the identity function and $r>0$?
Here, $B(f,r)=g in C[0,1]: textsup_x;vert x-g(x)vert<r. $
Is the figure look like the tube around the line $y=x$ ?
Added:
By DanielWainfleet answer, the figure look like this: (An outline with $r=1$)
Where the middle line is the identity function and and the upper line is the graph of $1+x$ and the bottom line is $-1+x$ .
A function $g in B(f,1)$ means $g$ lies entirely within this strip!
real-analysis metric-spaces
asked Aug 13 at 7:00
Learning Mathematics
543313
edited Aug 14 at 13:49
asked Aug 13 at 7:00
Learning Mathematics
543313
asked Aug 13 at 7:00
Learning Mathematics
543313
asked Aug 13 at 7:00
Learning Mathematics
543313
543313
1
Points in the ball are those functions whose graphs lie between the straight lines $y=x+r$ and $y =x-r$.
– Kavi Rama Murthy
Aug 13 at 7:20
add a comment |Â
1
Points in the ball are those functions whose graphs lie between the straight lines $y=x+r$ and $y =x-r$.
– Kavi Rama Murthy
Aug 13 at 7:20
1
1
Points in the ball are those functions whose graphs lie between the straight lines $y=x+r$ and $y =x-r$.
– Kavi Rama Murthy
Aug 13 at 7:20
Points in the ball are those functions whose graphs lie between the straight lines $y=x+r$ and $y =x-r$.
– Kavi Rama Murthy
Aug 13 at 7:20
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
3
down vote
accepted
Picture the graph of a continuous $f:0,1]to Bbb R$ and the graphs of $g(x)=r+f(x)$ and of $h(x)=-r+f(x).$ Continuous functions whose graphs lie entirely below $g$ and above $h$ are the members of $B(f,r) .$ In your Q a member $g$ of $B(id_[0,1],r)$ satisfies $-r+x<g(x)<r+x$ for every $x.$
BTW, this is sometimes called the metric of uniform convergence because $lim_nto inftyd_infty(f_n,f)=0$ iff the sequence $(f_n)_n$ of functions converges uniformly to $f.$
answered Aug 13 at 7:25
DanielWainfleet
31.9k31644
See my question again sir. I add the picture of the graph(the outline of the graph using mathematica) after understanding your thoughts! Is this correct?
– Learning Mathematics
Aug 14 at 13:43
1
Yes. That's exactly it.
– DanielWainfleet
Aug 15 at 23:22
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
Picture the graph of a continuous $f:0,1]to Bbb R$ and the graphs of $g(x)=r+f(x)$ and of $h(x)=-r+f(x).$ Continuous functions whose graphs lie entirely below $g$ and above $h$ are the members of $B(f,r) .$ In your Q a member $g$ of $B(id_[0,1],r)$ satisfies $-r+x<g(x)<r+x$ for every $x.$
BTW, this is sometimes called the metric of uniform convergence because $lim_nto inftyd_infty(f_n,f)=0$ iff the sequence $(f_n)_n$ of functions converges uniformly to $f.$
answered Aug 13 at 7:25
DanielWainfleet
31.9k31644
See my question again sir. I add the picture of the graph(the outline of the graph using mathematica) after understanding your thoughts! Is this correct?
– Learning Mathematics
Aug 14 at 13:43
1
Yes. That's exactly it.
– DanielWainfleet
Aug 15 at 23:22
add a comment |Â
up vote
3
down vote
accepted
Picture the graph of a continuous $f:0,1]to Bbb R$ and the graphs of $g(x)=r+f(x)$ and of $h(x)=-r+f(x).$ Continuous functions whose graphs lie entirely below $g$ and above $h$ are the members of $B(f,r) .$ In your Q a member $g$ of $B(id_[0,1],r)$ satisfies $-r+x<g(x)<r+x$ for every $x.$
BTW, this is sometimes called the metric of uniform convergence because $lim_nto inftyd_infty(f_n,f)=0$ iff the sequence $(f_n)_n$ of functions converges uniformly to $f.$
answered Aug 13 at 7:25
DanielWainfleet
31.9k31644
See my question again sir. I add the picture of the graph(the outline of the graph using mathematica) after understanding your thoughts! Is this correct?
– Learning Mathematics
Aug 14 at 13:43
1
Yes. That's exactly it.
– DanielWainfleet
Aug 15 at 23:22
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
Picture the graph of a continuous $f:0,1]to Bbb R$ and the graphs of $g(x)=r+f(x)$ and of $h(x)=-r+f(x).$ Continuous functions whose graphs lie entirely below $g$ and above $h$ are the members of $B(f,r) .$ In your Q a member $g$ of $B(id_[0,1],r)$ satisfies $-r+x<g(x)<r+x$ for every $x.$
BTW, this is sometimes called the metric of uniform convergence because $lim_nto inftyd_infty(f_n,f)=0$ iff the sequence $(f_n)_n$ of functions converges uniformly to $f.$
answered Aug 13 at 7:25
DanielWainfleet
31.9k31644
Picture the graph of a continuous $f:0,1]to Bbb R$ and the graphs of $g(x)=r+f(x)$ and of $h(x)=-r+f(x).$ Continuous functions whose graphs lie entirely below $g$ and above $h$ are the members of $B(f,r) .$ In your Q a member $g$ of $B(id_[0,1],r)$ satisfies $-r+x<g(x)<r+x$ for every $x.$
BTW, this is sometimes called the metric of uniform convergence because $lim_nto inftyd_infty(f_n,f)=0$ iff the sequence $(f_n)_n$ of functions converges uniformly to $f.$
answered Aug 13 at 7:25
DanielWainfleet
31.9k31644
answered Aug 13 at 7:25
DanielWainfleet
31.9k31644
answered Aug 13 at 7:25
DanielWainfleet
31.9k31644
answered Aug 13 at 7:25
DanielWainfleet
31.9k31644
31.9k31644
See my question again sir. I add the picture of the graph(the outline of the graph using mathematica) after understanding your thoughts! Is this correct?
– Learning Mathematics
Aug 14 at 13:43
1
Yes. That's exactly it.
– DanielWainfleet
Aug 15 at 23:22
add a comment |Â
See my question again sir. I add the picture of the graph(the outline of the graph using mathematica) after understanding your thoughts! Is this correct?
– Learning Mathematics
Aug 14 at 13:43
1
Yes. That's exactly it.
– DanielWainfleet
Aug 15 at 23:22
See my question again sir. I add the picture of the graph(the outline of the graph using mathematica) after understanding your thoughts! Is this correct?
– Learning Mathematics
Aug 14 at 13:43
See my question again sir. I add the picture of the graph(the outline of the graph using mathematica) after understanding your thoughts! Is this correct?
– Learning Mathematics
Aug 14 at 13:43
1
1
Yes. That's exactly it.
– DanielWainfleet
Aug 15 at 23:22
Yes. That's exactly it.
– DanielWainfleet
Aug 15 at 23:22
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2881083%2fhow-to-visualize-the-open-ball-in-c0-1%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
1
Points in the ball are those functions whose graphs lie between the straight lines $y=x+r$ and $y =x-r$.
– Kavi Rama Murthy
Aug 13 at 7:20