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Linear function, that is not continuous?
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Let $V:=fquadtextcontinuous$ provided with the norm $|f|_1:=int_0^1 |f(x)|, dx$ for $fin V$.
Show that $varphi : VtomathbbR$, $varphi(f):=f(0)$ is linear but not continuous.
It is easy to show, that $varphi$ is linear. Let $ainmathbbR$ and $f,gin V$. Then $varphi(af+g)=(af+g)(0)=af(0)+g(0)=avarphi(f)+varphi(g)$.
Now I want to show, that it is not linear.
I know some equivalent statements for a linear function $f:Vto W$ between two normed spaces, which I tried to disprove.
First, that there exists a $c>0$ such that $|f(v)|leq ccdot |v|$ for every $vin V$
I tried several sequences of functions, but they did not work out.
What I want to do is to give a sequence such that $f_n(0)=textconstant>0$ but $|f_n|_1to 0$ or that $f_n(0)to infty$ while $|f_n|_1$ stays bounded.
But nothing really worked. I tried a lot with $cos(nx)$ but the absolute value makes it hard to inegrate.
$cos$ or $sin$ should be involved I guess.
Hints are appreciated.
Thanks in advance.
functional-analysis continuity norm normed-spaces
asked Jul 18 at 18:11
Cornman
2,48921128
add a comment |Â
up vote
2
down vote
favorite
Let $V:=fquadtextcontinuous$ provided with the norm $|f|_1:=int_0^1 |f(x)|, dx$ for $fin V$.
Show that $varphi : VtomathbbR$, $varphi(f):=f(0)$ is linear but not continuous.
It is easy to show, that $varphi$ is linear. Let $ainmathbbR$ and $f,gin V$. Then $varphi(af+g)=(af+g)(0)=af(0)+g(0)=avarphi(f)+varphi(g)$.
Now I want to show, that it is not linear.
I know some equivalent statements for a linear function $f:Vto W$ between two normed spaces, which I tried to disprove.
First, that there exists a $c>0$ such that $|f(v)|leq ccdot |v|$ for every $vin V$
I tried several sequences of functions, but they did not work out.
What I want to do is to give a sequence such that $f_n(0)=textconstant>0$ but $|f_n|_1to 0$ or that $f_n(0)to infty$ while $|f_n|_1$ stays bounded.
But nothing really worked. I tried a lot with $cos(nx)$ but the absolute value makes it hard to inegrate.
$cos$ or $sin$ should be involved I guess.
Hints are appreciated.
Thanks in advance.
functional-analysis continuity norm normed-spaces
asked Jul 18 at 18:11
Cornman
2,48921128
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $V:=fquadtextcontinuous$ provided with the norm $|f|_1:=int_0^1 |f(x)|, dx$ for $fin V$.
Show that $varphi : VtomathbbR$, $varphi(f):=f(0)$ is linear but not continuous.
It is easy to show, that $varphi$ is linear. Let $ainmathbbR$ and $f,gin V$. Then $varphi(af+g)=(af+g)(0)=af(0)+g(0)=avarphi(f)+varphi(g)$.
Now I want to show, that it is not linear.
I know some equivalent statements for a linear function $f:Vto W$ between two normed spaces, which I tried to disprove.
First, that there exists a $c>0$ such that $|f(v)|leq ccdot |v|$ for every $vin V$
I tried several sequences of functions, but they did not work out.
What I want to do is to give a sequence such that $f_n(0)=textconstant>0$ but $|f_n|_1to 0$ or that $f_n(0)to infty$ while $|f_n|_1$ stays bounded.
But nothing really worked. I tried a lot with $cos(nx)$ but the absolute value makes it hard to inegrate.
$cos$ or $sin$ should be involved I guess.
Hints are appreciated.
Thanks in advance.
functional-analysis continuity norm normed-spaces
asked Jul 18 at 18:11
Cornman
2,48921128
Let $V:=fquadtextcontinuous$ provided with the norm $|f|_1:=int_0^1 |f(x)|, dx$ for $fin V$.
Show that $varphi : VtomathbbR$, $varphi(f):=f(0)$ is linear but not continuous.
It is easy to show, that $varphi$ is linear. Let $ainmathbbR$ and $f,gin V$. Then $varphi(af+g)=(af+g)(0)=af(0)+g(0)=avarphi(f)+varphi(g)$.
Now I want to show, that it is not linear.
I know some equivalent statements for a linear function $f:Vto W$ between two normed spaces, which I tried to disprove.
First, that there exists a $c>0$ such that $|f(v)|leq ccdot |v|$ for every $vin V$
I tried several sequences of functions, but they did not work out.
What I want to do is to give a sequence such that $f_n(0)=textconstant>0$ but $|f_n|_1to 0$ or that $f_n(0)to infty$ while $|f_n|_1$ stays bounded.
But nothing really worked. I tried a lot with $cos(nx)$ but the absolute value makes it hard to inegrate.
$cos$ or $sin$ should be involved I guess.
Hints are appreciated.
Thanks in advance.
functional-analysis continuity norm normed-spaces
asked Jul 18 at 18:11
Cornman
2,48921128
asked Jul 18 at 18:11
Cornman
2,48921128
asked Jul 18 at 18:11
Cornman
2,48921128
asked Jul 18 at 18:11
Cornman
2,48921128
2,48921128
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
Try with $f_n(t)=(1-t)^n$. The sequence $(f_n)_ninmathbb N$ converges to the null function, but you don't have $lim_ninmathbb Nvarphi(f_n)=varphi(0)=0$.
answered Jul 18 at 18:13
José Carlos Santos
114k1698177
2
Thank you. I feel so dumb right now, because my first try was $(1+x)^n$ but it did not work. And now a solution is just $(1-x)^n$. How could I not think about it... I tried at least one hour. Arrgh..
– Cornman
Jul 18 at 18:16
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Try with $f_n(t)=(1-t)^n$. The sequence $(f_n)_ninmathbb N$ converges to the null function, but you don't have $lim_ninmathbb Nvarphi(f_n)=varphi(0)=0$.
answered Jul 18 at 18:13
José Carlos Santos
114k1698177
2
Thank you. I feel so dumb right now, because my first try was $(1+x)^n$ but it did not work. And now a solution is just $(1-x)^n$. How could I not think about it... I tried at least one hour. Arrgh..
– Cornman
Jul 18 at 18:16
add a comment |Â
up vote
2
down vote
accepted
Try with $f_n(t)=(1-t)^n$. The sequence $(f_n)_ninmathbb N$ converges to the null function, but you don't have $lim_ninmathbb Nvarphi(f_n)=varphi(0)=0$.
answered Jul 18 at 18:13
José Carlos Santos
114k1698177
2
Thank you. I feel so dumb right now, because my first try was $(1+x)^n$ but it did not work. And now a solution is just $(1-x)^n$. How could I not think about it... I tried at least one hour. Arrgh..
– Cornman
Jul 18 at 18:16
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Try with $f_n(t)=(1-t)^n$. The sequence $(f_n)_ninmathbb N$ converges to the null function, but you don't have $lim_ninmathbb Nvarphi(f_n)=varphi(0)=0$.
answered Jul 18 at 18:13
José Carlos Santos
114k1698177
Try with $f_n(t)=(1-t)^n$. The sequence $(f_n)_ninmathbb N$ converges to the null function, but you don't have $lim_ninmathbb Nvarphi(f_n)=varphi(0)=0$.
answered Jul 18 at 18:13
José Carlos Santos
114k1698177
answered Jul 18 at 18:13
José Carlos Santos
114k1698177
answered Jul 18 at 18:13
José Carlos Santos
114k1698177
answered Jul 18 at 18:13
José Carlos Santos
114k1698177
114k1698177
2
Thank you. I feel so dumb right now, because my first try was $(1+x)^n$ but it did not work. And now a solution is just $(1-x)^n$. How could I not think about it... I tried at least one hour. Arrgh..
– Cornman
Jul 18 at 18:16
add a comment |Â
2
Thank you. I feel so dumb right now, because my first try was $(1+x)^n$ but it did not work. And now a solution is just $(1-x)^n$. How could I not think about it... I tried at least one hour. Arrgh..
– Cornman
Jul 18 at 18:16
2
2
Thank you. I feel so dumb right now, because my first try was $(1+x)^n$ but it did not work. And now a solution is just $(1-x)^n$. How could I not think about it... I tried at least one hour. Arrgh..
– Cornman
Jul 18 at 18:16
Thank you. I feel so dumb right now, because my first try was $(1+x)^n$ but it did not work. And now a solution is just $(1-x)^n$. How could I not think about it... I tried at least one hour. Arrgh..
– Cornman
Jul 18 at 18:16
add a comment |Â
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