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Thread: Image of bounded operator is closed

  1. #1
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    Image of bounded operator is closed

    Let $\displaystyle X, Y$ be Banach spaces, $\displaystyle U \in B(X,Y)$ be given such that $\displaystyle \exists M \in [0, \infty)$ such that $\displaystyle \forall y \in \text{Im}(U), \exists x \in X$ such that $\displaystyle U(x) = y$ and $\displaystyle \|x\|_X \leq M \|y\|_Y$. Show that $\displaystyle \text{Im}(U)$ is closed in $\displaystyle Y$.

    My idea was to use the quotient Banach space of $\displaystyle \frac{X}{\text{Ker}(U)}$ and use the First Isomorphism Theorem, but this isn't very direct. Is there a more direct method for proving this? Note that $\displaystyle U$ isn't necessarily injective.
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  2. #2
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    Quote Originally Posted by Giraffro View Post
    Let $\displaystyle X, Y$ be Banach spaces, $\displaystyle U \in B(X,Y)$ be given such that $\displaystyle \exists M \in [0, \infty)$ such that $\displaystyle \forall y \in \text{Im}(U), \exists x \in X$ such that $\displaystyle U(x) = y$ and $\displaystyle \|x\|_X \leq M \|y\|_Y$. Show that $\displaystyle \text{Im}(U)$ is closed in $\displaystyle Y$.

    My idea was to use the quotient Banach space of $\displaystyle \frac{X}{\text{Ker}(U)}$ and use the First Isomorphism Theorem, but this isn't very direct. Is there a more direct method for proving this? Note that $\displaystyle U$ isn't necessarily injective.
    Take a sequence y_n in the image, then let $\displaystyle U(x_n)=y_n$. Then

    $\displaystyle ||x_n-x_m|| \leq M ||U(x_n-x_m)||=||y_n-y_m|| \rightarrow 0$

    So that x_n is Cauchy and hence converges in X to some x. Now then by the continuity of U $\displaystyle y= \lim y_n= \lim U(x_n)= U(\lim x_n)=U(x)$ and hence y is in the image.
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  3. #3
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    Quote Originally Posted by Focus View Post
    Take a sequence y_n in the image, then let $\displaystyle U(x_n)=y_n$. Then

    $\displaystyle ||x_n-x_m|| \leq M ||U(x_n-x_m)||=||y_n-y_m|| \rightarrow 0$

    So that x_n is Cauchy and hence converges in X to some x. Now then by the continuity of U $\displaystyle y= \lim y_n= \lim U(x_n)= U(\lim x_n)=U(x)$ and hence y is in the image.
    That's wrong. You don't know that $\displaystyle \forall m,n \in \mathbb{N}, \|x_m - x_n\|_X \leq M \|U(x_m-x_n)\|_Y$. You can get that $\displaystyle \forall m, n \in \mathbb{N}, \exists x_{m,n} \in X$ such that $\displaystyle U(x_{m,n}) = y_m - y_n$ and $\displaystyle \|x_{m,n}\|_X \leq M \|y_m - y_n\|_Y$, but that's all.

    $\displaystyle x_{m,n}$ and $\displaystyle x_m - x_n$ can differ by anything in $\displaystyle \text{Ker}(U)$, which is why I used the quotient norm, but there has to be a more direct method.
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  4. #4
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    Quote Originally Posted by Giraffro View Post
    That's wrong. You don't know that $\displaystyle \forall m,n \in \mathbb{N}, \|x_m - x_n\|_X \leq M \|U(x_m-x_n)\|_Y$. You can get that $\displaystyle \forall m, n \in \mathbb{N}, \exists x_{m,n} \in X$ such that $\displaystyle U(x_{m,n}) = y_m - y_n$ and $\displaystyle \|x_{m,n}\|_X \leq M \|y_m - y_n\|_Y$, but that's all.

    $\displaystyle x_{m,n}$ and $\displaystyle x_m - x_n$ can differ by anything in $\displaystyle \text{Ker}(U)$, which is why I used the quotient norm, but there has to be a more direct method.
    I think that quotienting by ker(U) is probably essential, because it seems to be the only way to get round the difficulty described above. If you then consider the induced map $\displaystyle \hat{U}:X/\ker(U)\to Y$, it is injective and still satisfies the condition $\displaystyle \|\hat{x}\|_{X/\ker(U)}\leqslant M\|\hat{U}(\hat{x})\|_Y$, and so the argument suggested by Focus becomes valid.
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