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Qingping Zeng
Qingping Zeng
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Let $X$ and $Y$ be Banach spaces. The notion of the embedded spaces was introduced by D.S. Djordjevic. Say that $X$ embed in $Y$, and write $X \preceq Y$, if there exists a left invertible operator $J:X \rightarrow Y$. My question is the following；

If $X$ embed in $Y$, and $Y$ embed in $X$, then $X$ isomorphic onto $Y$？

PS: The answderanswer is positive when $X$ and $Y$ are Hilbert spaces. But for general Banach space,I can not find a solution. Is there exists a couterexample?

Let $X$ and $Y$ be Banach spaces. The notion of the embedded spaces was introduced by D.S. Djordjevic. Say that $X$ embed in $Y$, and write $X \preceq Y$, if there exists a left invertible operator $J:X \rightarrow Y$. My question is the following；

If $X$ embed in $Y$, and $Y$ embed in $X$, then $X$ isomorphic onto $Y$？

PS: The answder is positive when $X$ and $Y$ are Hilbert spaces. But for general Banach space,I can not find a solution. Is there exists a couterexample?

Let $X$ and $Y$ be Banach spaces. The notion of the embedded spaces was introduced by D.S. Djordjevic. Say that $X$ embed in $Y$, and write $X \preceq Y$, if there exists a left invertible operator $J:X \rightarrow Y$. My question is the following；

If $X$ embed in $Y$, and $Y$ embed in $X$, then $X$ isomorphic onto $Y$？

PS: The answer is positive when $X$ and $Y$ are Hilbert spaces. But for general Banach space,I can not find a solution. Is there exists a couterexample?

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Qingping Zeng
Qingping Zeng
  • 515
  • 2
  • 8

Does X$X$ embed in Y$Y$, and Y$Y$ embed in X$X$, always imply that X$X$ isomorphic (on)to Yonto $Y$?

Let X$X$ and Y$Y$ be Banach spaces. The notion of the embedded spaces was introduced by D.S. Djordjevic. Say that X$X$ embed in Y$Y$, and write X \preceq Y$X \preceq Y$, if there exists a left invertible operator J:X \rightarrow Y$J:X \rightarrow Y$. My question is the following；

If X$X$ embed in Y$Y$, and Y$Y$ embed in X$X$, then X$X$ isomorphic onto (on)to Y？$Y$？

PS: The answder is positive when X$X$ and Y$Y$ are Hilbert spaces. But for general Banach space,I can not find a solution. Is there exists a couterexample?

Does X embed in Y, and Y embed in X, always imply that X isomorphic (on)to Y

Let X and Y be Banach spaces. The notion of the embedded spaces was introduced by D.S. Djordjevic. Say that X embed in Y, and write X \preceq Y, if there exists a left invertible operator J:X \rightarrow Y. My question is the following；

If X embed in Y, and Y embed in X, then X isomorphic (on)to Y？

PS: The answder is positive when X and Y are Hilbert spaces. But for general Banach space,I can not find a solution. Is there exists a couterexample?

Does $X$ embed in $Y$, and $Y$ embed in $X$, always imply that $X$ isomorphic onto $Y$?

Let $X$ and $Y$ be Banach spaces. The notion of the embedded spaces was introduced by D.S. Djordjevic. Say that $X$ embed in $Y$, and write $X \preceq Y$, if there exists a left invertible operator $J:X \rightarrow Y$. My question is the following；

If $X$ embed in $Y$, and $Y$ embed in $X$, then $X$ isomorphic onto $Y$？

PS: The answder is positive when $X$ and $Y$ are Hilbert spaces. But for general Banach space,I can not find a solution. Is there exists a couterexample?

Source Link
Qingping Zeng
Qingping Zeng
  • 515
  • 2
  • 8

Does X embed in Y, and Y embed in X, always imply that X isomorphic (on)to Y

Let X and Y be Banach spaces. The notion of the embedded spaces was introduced by D.S. Djordjevic. Say that X embed in Y, and write X \preceq Y, if there exists a left invertible operator J:X \rightarrow Y. My question is the following；

If X embed in Y, and Y embed in X, then X isomorphic (on)to Y？

PS: The answder is positive when X and Y are Hilbert spaces. But for general Banach space,I can not find a solution. Is there exists a couterexample?