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Let the space be $\mathcal{F} = \{g(x) = \int_{-c}^c h(w)e^{(\sigma+i w)x}: h(w)\in L^2[-c,c], x\in[0,T]\}$. I want to prove the space is Banach space with the $l_2-$norm $\|g\|_2$ or $l_\infty$-norm $\|g\| = \max_{t\in[0,T]}|g(x)|$.

Reason: Since the space $\mathcal{F}$ is subspace of $L^2[0,T]$ or $L^\infty[0,T]$, I want to show that it is closed with $l^2$ norm, or $l^\infty$ norm. But I do not know how to prove it.

It is really important to me. Thanks a lot!

xuanxuan
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  • If you look at your set $\mathcal{F}$ as image of a linear operator $A$ from $L^2([-c,c])$ to $L^2([0,T])$ and you decompose such operator as composition of several maps you might end up with the answer (at least for the $L^2$ case). Using the definition of completeness does not seem friendly here. – user8469759 Sep 20 '22 at 16:59
  • @user8469759 Thank you for your comment! Do you mean that consider this problem of closed space by closed linear operator ? It would be really helpful if you explain more about the decomposition of such operator as composition of several maps. Because I still do not know how to prove this problem for $L^2$ case by the decomposition. In particular, what theorems is the decomposition based on? How to prove the closedness of operator by decomposition? – xuanxuan Sep 21 '22 at 03:32
  • @user8469759 Oh! I think I might understand the decomposition. First, from $L^2[-c,c]\to L^2[-\infty,\infty]$ (Inverse Fourier transform), we get $g_1(x) = \int_{-c}^c h(w)e^{iwx}dx,x\in R$. Second, from $L^2[-\infty,\infty]$ to $L^2[-0,T]$, we get $g_1(x) = \int_{-c}^c h(w)e^{iwx}dx,x\in [0,T]$. Finally, we multiply $e^{\sigma x}$ But I find that the problem is whether the second operator, from $L^2[-\infty,\infty]\to L^2[0,T]$, can be closed. Do you know how to prove that there is a $T>0$ and $a>0$ such that $\frac{\int_{0}^T|g_2(x)|^2dx}{\int_{-\infty}^{\infty}|g_1(x)|^2 dx}\ge a$? – xuanxuan Sep 21 '22 at 06:54
  • If you want we can have a chat, cause I find your problem interesting (and whatever I have in mind sounds long for a comment but not complete enough for an answer). However I don't seem to be able to invite you. I think you're on the right track anyway but I didn't get your last statement. – user8469759 Sep 21 '22 at 10:11
  • @user8469759 Sure! My email address is xuanxuan@hust.edu.cn. My last statement is to prove the related operator is closed, which is based on the theorem in link – xuanxuan Sep 21 '22 at 16:01
  • Try to join here https://chat.stackexchange.com/rooms/139368/complete-space-with-fourier-operator – user8469759 Sep 22 '22 at 10:06

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