Image of a transformation becomes a Hilbert space !!












0












$begingroup$


Let $T: L^2(mathbb R)to O(mathbb C)$ an transformation from the Hilbert space $L^2(mathbb R)$ to the Fréchet spaces $O(mathbb C)$ of all holomorphic functions on $mathbb C$. We assume that $T$ is injective and continuous.



Now, Can we endowed the space $mathcal H:= Im T$ (image of $T$) with the Hilbert inner product
from $left(L^{2}(mathbb{R}), left<.,.right>right)$ for that $mathcal H:= Im T$ becomes a Hilbert space of holomorphic functions ? If yes, how?



I think it is possible if we consider the inner product $(f,g)_{mathcal H}=left<T^{-1}(f),T^{-1}(g)right>_{L^{2}(mathbb{R})}$ ??



Thank you in advance






complex-analysis operator-theory hilbert-spaces harmonic-analysis







share|cite|improve this question











$endgroup$












  • $begingroup$
    I suspect that you need the inverse of $T$ to be continuous in order to guarantee completeness. This is not the case in general.
    $endgroup$
    – Martin
    Dec 3 '18 at 16:29










  • $begingroup$
    @ Martin, we have assume that $T$ is injective and continuous. Then, $T$ becomes an isomorphism continuous from $L^{2}(mathbb{R})$ to $Im T$.
    $endgroup$
    – Z. Alfata
    Dec 3 '18 at 16:53










  • $begingroup$
    Yes but take a Cauchy sequence in the image. You now need to show that it's convergent. Now the you'd like to map that Cauchy sequence back to $L^2$ in order to use completeness of this space. However we hit a wall - the inverse is not necessarily continuous - thus elements close to each other in $H$ need not be close in $L^2$. I'm not saying it's not doable just that we might need continuity of $T^{-1}$ that's all :)
    $endgroup$
    – Martin
    Dec 3 '18 at 17:00










  • $begingroup$
    @ Martin, I think it is possible if we consider the inner product $(f,g)_{mathcal H}=left<T^{-1}(f),T^{-1}(g)right>_{L^{2}(mathbb{R})}$ ??
    $endgroup$
    – Z. Alfata
    Dec 5 '18 at 9:41
















0












$begingroup$


Let $T: L^2(mathbb R)to O(mathbb C)$ an transformation from the Hilbert space $L^2(mathbb R)$ to the Fréchet spaces $O(mathbb C)$ of all holomorphic functions on $mathbb C$. We assume that $T$ is injective and continuous.



Now, Can we endowed the space $mathcal H:= Im T$ (image of $T$) with the Hilbert inner product
from $left(L^{2}(mathbb{R}), left<.,.right>right)$ for that $mathcal H:= Im T$ becomes a Hilbert space of holomorphic functions ? If yes, how?



I think it is possible if we consider the inner product $(f,g)_{mathcal H}=left<T^{-1}(f),T^{-1}(g)right>_{L^{2}(mathbb{R})}$ ??



Thank you in advance






complex-analysis operator-theory hilbert-spaces harmonic-analysis







share|cite|improve this question











$endgroup$












  • $begingroup$
    I suspect that you need the inverse of $T$ to be continuous in order to guarantee completeness. This is not the case in general.
    $endgroup$
    – Martin
    Dec 3 '18 at 16:29










  • $begingroup$
    @ Martin, we have assume that $T$ is injective and continuous. Then, $T$ becomes an isomorphism continuous from $L^{2}(mathbb{R})$ to $Im T$.
    $endgroup$
    – Z. Alfata
    Dec 3 '18 at 16:53










  • $begingroup$
    Yes but take a Cauchy sequence in the image. You now need to show that it's convergent. Now the you'd like to map that Cauchy sequence back to $L^2$ in order to use completeness of this space. However we hit a wall - the inverse is not necessarily continuous - thus elements close to each other in $H$ need not be close in $L^2$. I'm not saying it's not doable just that we might need continuity of $T^{-1}$ that's all :)
    $endgroup$
    – Martin
    Dec 3 '18 at 17:00










  • $begingroup$
    @ Martin, I think it is possible if we consider the inner product $(f,g)_{mathcal H}=left<T^{-1}(f),T^{-1}(g)right>_{L^{2}(mathbb{R})}$ ??
    $endgroup$
    – Z. Alfata
    Dec 5 '18 at 9:41














0












0








0





$begingroup$


Let $T: L^2(mathbb R)to O(mathbb C)$ an transformation from the Hilbert space $L^2(mathbb R)$ to the Fréchet spaces $O(mathbb C)$ of all holomorphic functions on $mathbb C$. We assume that $T$ is injective and continuous.



Now, Can we endowed the space $mathcal H:= Im T$ (image of $T$) with the Hilbert inner product
from $left(L^{2}(mathbb{R}), left<.,.right>right)$ for that $mathcal H:= Im T$ becomes a Hilbert space of holomorphic functions ? If yes, how?



I think it is possible if we consider the inner product $(f,g)_{mathcal H}=left<T^{-1}(f),T^{-1}(g)right>_{L^{2}(mathbb{R})}$ ??



Thank you in advance






complex-analysis operator-theory hilbert-spaces harmonic-analysis







share|cite|improve this question











$endgroup$




Let $T: L^2(mathbb R)to O(mathbb C)$ an transformation from the Hilbert space $L^2(mathbb R)$ to the Fréchet spaces $O(mathbb C)$ of all holomorphic functions on $mathbb C$. We assume that $T$ is injective and continuous.



Now, Can we endowed the space $mathcal H:= Im T$ (image of $T$) with the Hilbert inner product
from $left(L^{2}(mathbb{R}), left<.,.right>right)$ for that $mathcal H:= Im T$ becomes a Hilbert space of holomorphic functions ? If yes, how?



I think it is possible if we consider the inner product $(f,g)_{mathcal H}=left<T^{-1}(f),T^{-1}(g)right>_{L^{2}(mathbb{R})}$ ??



Thank you in advance






complex-analysis operator-theory hilbert-spaces harmonic-analysis




complex-analysis operator-theory hilbert-spaces harmonic-analysis






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 5 '18 at 9:40







Z. Alfata

















asked Dec 3 '18 at 15:25









Z. AlfataZ. Alfata

895514




895514












  • $begingroup$
    I suspect that you need the inverse of $T$ to be continuous in order to guarantee completeness. This is not the case in general.
    $endgroup$
    – Martin
    Dec 3 '18 at 16:29










  • $begingroup$
    @ Martin, we have assume that $T$ is injective and continuous. Then, $T$ becomes an isomorphism continuous from $L^{2}(mathbb{R})$ to $Im T$.
    $endgroup$
    – Z. Alfata
    Dec 3 '18 at 16:53










  • $begingroup$
    Yes but take a Cauchy sequence in the image. You now need to show that it's convergent. Now the you'd like to map that Cauchy sequence back to $L^2$ in order to use completeness of this space. However we hit a wall - the inverse is not necessarily continuous - thus elements close to each other in $H$ need not be close in $L^2$. I'm not saying it's not doable just that we might need continuity of $T^{-1}$ that's all :)
    $endgroup$
    – Martin
    Dec 3 '18 at 17:00










  • $begingroup$
    @ Martin, I think it is possible if we consider the inner product $(f,g)_{mathcal H}=left<T^{-1}(f),T^{-1}(g)right>_{L^{2}(mathbb{R})}$ ??
    $endgroup$
    – Z. Alfata
    Dec 5 '18 at 9:41


















  • $begingroup$
    I suspect that you need the inverse of $T$ to be continuous in order to guarantee completeness. This is not the case in general.
    $endgroup$
    – Martin
    Dec 3 '18 at 16:29










  • $begingroup$
    @ Martin, we have assume that $T$ is injective and continuous. Then, $T$ becomes an isomorphism continuous from $L^{2}(mathbb{R})$ to $Im T$.
    $endgroup$
    – Z. Alfata
    Dec 3 '18 at 16:53










  • $begingroup$
    Yes but take a Cauchy sequence in the image. You now need to show that it's convergent. Now the you'd like to map that Cauchy sequence back to $L^2$ in order to use completeness of this space. However we hit a wall - the inverse is not necessarily continuous - thus elements close to each other in $H$ need not be close in $L^2$. I'm not saying it's not doable just that we might need continuity of $T^{-1}$ that's all :)
    $endgroup$
    – Martin
    Dec 3 '18 at 17:00










  • $begingroup$
    @ Martin, I think it is possible if we consider the inner product $(f,g)_{mathcal H}=left<T^{-1}(f),T^{-1}(g)right>_{L^{2}(mathbb{R})}$ ??
    $endgroup$
    – Z. Alfata
    Dec 5 '18 at 9:41
















$begingroup$
I suspect that you need the inverse of $T$ to be continuous in order to guarantee completeness. This is not the case in general.
$endgroup$
– Martin
Dec 3 '18 at 16:29




$begingroup$
I suspect that you need the inverse of $T$ to be continuous in order to guarantee completeness. This is not the case in general.
$endgroup$
– Martin
Dec 3 '18 at 16:29












$begingroup$
@ Martin, we have assume that $T$ is injective and continuous. Then, $T$ becomes an isomorphism continuous from $L^{2}(mathbb{R})$ to $Im T$.
$endgroup$
– Z. Alfata
Dec 3 '18 at 16:53




$begingroup$
@ Martin, we have assume that $T$ is injective and continuous. Then, $T$ becomes an isomorphism continuous from $L^{2}(mathbb{R})$ to $Im T$.
$endgroup$
– Z. Alfata
Dec 3 '18 at 16:53












$begingroup$
Yes but take a Cauchy sequence in the image. You now need to show that it's convergent. Now the you'd like to map that Cauchy sequence back to $L^2$ in order to use completeness of this space. However we hit a wall - the inverse is not necessarily continuous - thus elements close to each other in $H$ need not be close in $L^2$. I'm not saying it's not doable just that we might need continuity of $T^{-1}$ that's all :)
$endgroup$
– Martin
Dec 3 '18 at 17:00




$begingroup$
Yes but take a Cauchy sequence in the image. You now need to show that it's convergent. Now the you'd like to map that Cauchy sequence back to $L^2$ in order to use completeness of this space. However we hit a wall - the inverse is not necessarily continuous - thus elements close to each other in $H$ need not be close in $L^2$. I'm not saying it's not doable just that we might need continuity of $T^{-1}$ that's all :)
$endgroup$
– Martin
Dec 3 '18 at 17:00












$begingroup$
@ Martin, I think it is possible if we consider the inner product $(f,g)_{mathcal H}=left<T^{-1}(f),T^{-1}(g)right>_{L^{2}(mathbb{R})}$ ??
$endgroup$
– Z. Alfata
Dec 5 '18 at 9:41




$begingroup$
@ Martin, I think it is possible if we consider the inner product $(f,g)_{mathcal H}=left<T^{-1}(f),T^{-1}(g)right>_{L^{2}(mathbb{R})}$ ??
$endgroup$
– Z. Alfata
Dec 5 '18 at 9:41










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