Equivalence of injectivity of a lineal operator $T$ between two Banach spaces and and existence of another...












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Given $X$ and $Y$ two Banach spaces and $T in L(X,Y)$, where $L(X,Y)$ is the set of all the linear operators of $X$ in $Y$, show that the following sentences are equivalent:



a) There exists $S in L(X,Y)$ such that $S(T(x))=x , , forall x in X$.



b) $T$ is injective and $T(X)$ is a complemented space of $Y$.






functional-analysis linear-transformations banach-spaces







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    What where your attempts in proving this?
    – James
    yesterday
















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Given $X$ and $Y$ two Banach spaces and $T in L(X,Y)$, where $L(X,Y)$ is the set of all the linear operators of $X$ in $Y$, show that the following sentences are equivalent:



a) There exists $S in L(X,Y)$ such that $S(T(x))=x , , forall x in X$.



b) $T$ is injective and $T(X)$ is a complemented space of $Y$.






functional-analysis linear-transformations banach-spaces







share|cite|improve this question


















  • 1




    What where your attempts in proving this?
    – James
    yesterday














0












0








0


1





Given $X$ and $Y$ two Banach spaces and $T in L(X,Y)$, where $L(X,Y)$ is the set of all the linear operators of $X$ in $Y$, show that the following sentences are equivalent:



a) There exists $S in L(X,Y)$ such that $S(T(x))=x , , forall x in X$.



b) $T$ is injective and $T(X)$ is a complemented space of $Y$.






functional-analysis linear-transformations banach-spaces







share|cite|improve this question













Given $X$ and $Y$ two Banach spaces and $T in L(X,Y)$, where $L(X,Y)$ is the set of all the linear operators of $X$ in $Y$, show that the following sentences are equivalent:



a) There exists $S in L(X,Y)$ such that $S(T(x))=x , , forall x in X$.



b) $T$ is injective and $T(X)$ is a complemented space of $Y$.






functional-analysis linear-transformations banach-spaces




functional-analysis linear-transformations banach-spaces






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Maggie94

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  • 1




    What where your attempts in proving this?
    – James
    yesterday














  • 1




    What where your attempts in proving this?
    – James
    yesterday








1




1




What where your attempts in proving this?
– James
yesterday




What where your attempts in proving this?
– James
yesterday










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Your $S$ should be a linear operator on $Y$, otherwise the composition makes no sense. The direction a) to b) is simple, just work with the definition of an injective map -- you don't need anything else. If you can't manage, I will add a proof here but at least try to solve it yourself.



For b) implies a), let ${x_imid iin I}$ be a basis for $X$. By infectivity of $T$, ${T(x_i)mid iin I}$ forms a basis of the image of $T$ -- convince yourself that this is true and let me know if you can't. Now define $$S:Yto X, smapstobegin{cases}T^{-1}(s),& if;sinmathrm{img}(T),\ 0,&else.end{cases}$$ You can then check by writing $s=sum alpha_ix_i$ that $S$ is as you want it to. Can you fill the missing details yourself?






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    Your $S$ should be a linear operator on $Y$, otherwise the composition makes no sense. The direction a) to b) is simple, just work with the definition of an injective map -- you don't need anything else. If you can't manage, I will add a proof here but at least try to solve it yourself.



    For b) implies a), let ${x_imid iin I}$ be a basis for $X$. By infectivity of $T$, ${T(x_i)mid iin I}$ forms a basis of the image of $T$ -- convince yourself that this is true and let me know if you can't. Now define $$S:Yto X, smapstobegin{cases}T^{-1}(s),& if;sinmathrm{img}(T),\ 0,&else.end{cases}$$ You can then check by writing $s=sum alpha_ix_i$ that $S$ is as you want it to. Can you fill the missing details yourself?






    share|cite|improve this answer


























      0














      Your $S$ should be a linear operator on $Y$, otherwise the composition makes no sense. The direction a) to b) is simple, just work with the definition of an injective map -- you don't need anything else. If you can't manage, I will add a proof here but at least try to solve it yourself.



      For b) implies a), let ${x_imid iin I}$ be a basis for $X$. By infectivity of $T$, ${T(x_i)mid iin I}$ forms a basis of the image of $T$ -- convince yourself that this is true and let me know if you can't. Now define $$S:Yto X, smapstobegin{cases}T^{-1}(s),& if;sinmathrm{img}(T),\ 0,&else.end{cases}$$ You can then check by writing $s=sum alpha_ix_i$ that $S$ is as you want it to. Can you fill the missing details yourself?






      share|cite|improve this answer
























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        Your $S$ should be a linear operator on $Y$, otherwise the composition makes no sense. The direction a) to b) is simple, just work with the definition of an injective map -- you don't need anything else. If you can't manage, I will add a proof here but at least try to solve it yourself.



        For b) implies a), let ${x_imid iin I}$ be a basis for $X$. By infectivity of $T$, ${T(x_i)mid iin I}$ forms a basis of the image of $T$ -- convince yourself that this is true and let me know if you can't. Now define $$S:Yto X, smapstobegin{cases}T^{-1}(s),& if;sinmathrm{img}(T),\ 0,&else.end{cases}$$ You can then check by writing $s=sum alpha_ix_i$ that $S$ is as you want it to. Can you fill the missing details yourself?






        share|cite|improve this answer












        Your $S$ should be a linear operator on $Y$, otherwise the composition makes no sense. The direction a) to b) is simple, just work with the definition of an injective map -- you don't need anything else. If you can't manage, I will add a proof here but at least try to solve it yourself.



        For b) implies a), let ${x_imid iin I}$ be a basis for $X$. By infectivity of $T$, ${T(x_i)mid iin I}$ forms a basis of the image of $T$ -- convince yourself that this is true and let me know if you can't. Now define $$S:Yto X, smapstobegin{cases}T^{-1}(s),& if;sinmathrm{img}(T),\ 0,&else.end{cases}$$ You can then check by writing $s=sum alpha_ix_i$ that $S$ is as you want it to. Can you fill the missing details yourself?







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










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        James

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