Question about inner product of $C_{2}[a,b]$.












1












$begingroup$


My question, I think, is quite simple.
I have a space $C_{2}[a,b]$ of complex-valued continuous functions. I checked that functional $$(f,g) = int_{a}^{b} f(t)overline{g}(t)dt$$ is a inner product on this space.
But I don't know how should I work with them.
For example, I have a function $g(t) = t^3$ hence $g(t) = (x+iy)^3$ and $overline{g}(t) = (x-iy)^3$?
I should consider any function $u(x,y)$ as sum: $u(x,y) = f(x,y) + ih(x,y)$?
Thank you very much and sorry for pretty elementary question..






linear-algebra functional-analysis hilbert-spaces







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$endgroup$








  • 2




    $begingroup$
    For $g:tmapsto t^3$ for example, you would have : $||g||^2=(g,g)=int_{a}^{b}t^3bar{t}^3dt=int_{a}^{b}t^6dt=frac{1}{7}(b^7-a^7). $ The variable lies in $[a;b]$. It's the images that can be complex.
    $endgroup$
    – Ayoub
    Jan 20 at 13:49












  • $begingroup$
    Thank you very much!
    $endgroup$
    – mathmaniac
    Jan 20 at 14:06
















1












$begingroup$


My question, I think, is quite simple.
I have a space $C_{2}[a,b]$ of complex-valued continuous functions. I checked that functional $$(f,g) = int_{a}^{b} f(t)overline{g}(t)dt$$ is a inner product on this space.
But I don't know how should I work with them.
For example, I have a function $g(t) = t^3$ hence $g(t) = (x+iy)^3$ and $overline{g}(t) = (x-iy)^3$?
I should consider any function $u(x,y)$ as sum: $u(x,y) = f(x,y) + ih(x,y)$?
Thank you very much and sorry for pretty elementary question..






linear-algebra functional-analysis hilbert-spaces







share|cite|improve this question









$endgroup$








  • 2




    $begingroup$
    For $g:tmapsto t^3$ for example, you would have : $||g||^2=(g,g)=int_{a}^{b}t^3bar{t}^3dt=int_{a}^{b}t^6dt=frac{1}{7}(b^7-a^7). $ The variable lies in $[a;b]$. It's the images that can be complex.
    $endgroup$
    – Ayoub
    Jan 20 at 13:49












  • $begingroup$
    Thank you very much!
    $endgroup$
    – mathmaniac
    Jan 20 at 14:06














1












1








1





$begingroup$


My question, I think, is quite simple.
I have a space $C_{2}[a,b]$ of complex-valued continuous functions. I checked that functional $$(f,g) = int_{a}^{b} f(t)overline{g}(t)dt$$ is a inner product on this space.
But I don't know how should I work with them.
For example, I have a function $g(t) = t^3$ hence $g(t) = (x+iy)^3$ and $overline{g}(t) = (x-iy)^3$?
I should consider any function $u(x,y)$ as sum: $u(x,y) = f(x,y) + ih(x,y)$?
Thank you very much and sorry for pretty elementary question..






linear-algebra functional-analysis hilbert-spaces







share|cite|improve this question









$endgroup$




My question, I think, is quite simple.
I have a space $C_{2}[a,b]$ of complex-valued continuous functions. I checked that functional $$(f,g) = int_{a}^{b} f(t)overline{g}(t)dt$$ is a inner product on this space.
But I don't know how should I work with them.
For example, I have a function $g(t) = t^3$ hence $g(t) = (x+iy)^3$ and $overline{g}(t) = (x-iy)^3$?
I should consider any function $u(x,y)$ as sum: $u(x,y) = f(x,y) + ih(x,y)$?
Thank you very much and sorry for pretty elementary question..






linear-algebra functional-analysis hilbert-spaces




linear-algebra functional-analysis hilbert-spaces






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 20 at 13:40









mathmaniacmathmaniac

19012




19012








  • 2




    $begingroup$
    For $g:tmapsto t^3$ for example, you would have : $||g||^2=(g,g)=int_{a}^{b}t^3bar{t}^3dt=int_{a}^{b}t^6dt=frac{1}{7}(b^7-a^7). $ The variable lies in $[a;b]$. It's the images that can be complex.
    $endgroup$
    – Ayoub
    Jan 20 at 13:49












  • $begingroup$
    Thank you very much!
    $endgroup$
    – mathmaniac
    Jan 20 at 14:06














  • 2




    $begingroup$
    For $g:tmapsto t^3$ for example, you would have : $||g||^2=(g,g)=int_{a}^{b}t^3bar{t}^3dt=int_{a}^{b}t^6dt=frac{1}{7}(b^7-a^7). $ The variable lies in $[a;b]$. It's the images that can be complex.
    $endgroup$
    – Ayoub
    Jan 20 at 13:49












  • $begingroup$
    Thank you very much!
    $endgroup$
    – mathmaniac
    Jan 20 at 14:06








2




2




$begingroup$
For $g:tmapsto t^3$ for example, you would have : $||g||^2=(g,g)=int_{a}^{b}t^3bar{t}^3dt=int_{a}^{b}t^6dt=frac{1}{7}(b^7-a^7). $ The variable lies in $[a;b]$. It's the images that can be complex.
$endgroup$
– Ayoub
Jan 20 at 13:49






$begingroup$
For $g:tmapsto t^3$ for example, you would have : $||g||^2=(g,g)=int_{a}^{b}t^3bar{t}^3dt=int_{a}^{b}t^6dt=frac{1}{7}(b^7-a^7). $ The variable lies in $[a;b]$. It's the images that can be complex.
$endgroup$
– Ayoub
Jan 20 at 13:49














$begingroup$
Thank you very much!
$endgroup$
– mathmaniac
Jan 20 at 14:06




$begingroup$
Thank you very much!
$endgroup$
– mathmaniac
Jan 20 at 14:06










1 Answer
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$begingroup$

A function $f:[a,b]tomathbb C$ is of the form $f(t)=f_1(t)+if_2(t)$ where $f_1$ and $f_2$ are real valued functions.



In particular, for $g(t)=t^3$, this is just a real valued function.






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Thank you very much!
    $endgroup$
    – mathmaniac
    Jan 20 at 14:06










  • $begingroup$
    It's a pleasure :)
    $endgroup$
    – Scientifica
    Jan 20 at 14:06











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1 Answer
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1 Answer
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active

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active

oldest

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active

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1












$begingroup$

A function $f:[a,b]tomathbb C$ is of the form $f(t)=f_1(t)+if_2(t)$ where $f_1$ and $f_2$ are real valued functions.



In particular, for $g(t)=t^3$, this is just a real valued function.






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Thank you very much!
    $endgroup$
    – mathmaniac
    Jan 20 at 14:06










  • $begingroup$
    It's a pleasure :)
    $endgroup$
    – Scientifica
    Jan 20 at 14:06
















1












$begingroup$

A function $f:[a,b]tomathbb C$ is of the form $f(t)=f_1(t)+if_2(t)$ where $f_1$ and $f_2$ are real valued functions.



In particular, for $g(t)=t^3$, this is just a real valued function.






share|cite|improve this answer









$endgroup$









  • 1




    $begingroup$
    Thank you very much!
    $endgroup$
    – mathmaniac
    Jan 20 at 14:06










  • $begingroup$
    It's a pleasure :)
    $endgroup$
    – Scientifica
    Jan 20 at 14:06














1












1








1





$begingroup$

A function $f:[a,b]tomathbb C$ is of the form $f(t)=f_1(t)+if_2(t)$ where $f_1$ and $f_2$ are real valued functions.



In particular, for $g(t)=t^3$, this is just a real valued function.






share|cite|improve this answer









$endgroup$



A function $f:[a,b]tomathbb C$ is of the form $f(t)=f_1(t)+if_2(t)$ where $f_1$ and $f_2$ are real valued functions.



In particular, for $g(t)=t^3$, this is just a real valued function.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Jan 20 at 13:53









ScientificaScientifica

6,79641335




6,79641335








  • 1




    $begingroup$
    Thank you very much!
    $endgroup$
    – mathmaniac
    Jan 20 at 14:06










  • $begingroup$
    It's a pleasure :)
    $endgroup$
    – Scientifica
    Jan 20 at 14:06














  • 1




    $begingroup$
    Thank you very much!
    $endgroup$
    – mathmaniac
    Jan 20 at 14:06










  • $begingroup$
    It's a pleasure :)
    $endgroup$
    – Scientifica
    Jan 20 at 14:06








1




1




$begingroup$
Thank you very much!
$endgroup$
– mathmaniac
Jan 20 at 14:06




$begingroup$
Thank you very much!
$endgroup$
– mathmaniac
Jan 20 at 14:06












$begingroup$
It's a pleasure :)
$endgroup$
– Scientifica
Jan 20 at 14:06




$begingroup$
It's a pleasure :)
$endgroup$
– Scientifica
Jan 20 at 14:06


















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