Hilbert manifold decomposition into infinite-dimensional ellipsoids

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Let $X$ be a Hilbert manifold and $mathcalU_alpha$ an open subet of $X$, with a local coordinate chart $(mathcalU_alpha,phi_alpha)$ such that $X:=bigcup_alphain AmathcalU_alpha.$ Suppose the chart is given by $$phi_alpha:mathcalU_alphatomathbbR^infty \ gamma_alphamapsto (phi_alpha^1,phi_alpha^2,...).$$ Also suppose the local coordinates satisfy $$sum_k=1^inftyc_k^2||phi_alpha^k||_X^2=R_alpha^2$$ for $c_ine 0,i=1,...$.




Then is $X$ the union of infinite-dimensional ellipsoids embedded in $mathbbR^infty$ over $alphain A$?




Thanks in advance!




hilbert-spaces conic-sections




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    Let $X$ be a Hilbert manifold and $mathcalU_alpha$ an open subet of $X$, with a local coordinate chart $(mathcalU_alpha,phi_alpha)$ such that $X:=bigcup_alphain AmathcalU_alpha.$ Suppose the chart is given by $$phi_alpha:mathcalU_alphatomathbbR^infty \ gamma_alphamapsto (phi_alpha^1,phi_alpha^2,...).$$ Also suppose the local coordinates satisfy $$sum_k=1^inftyc_k^2||phi_alpha^k||_X^2=R_alpha^2$$ for $c_ine 0,i=1,...$.




    Then is $X$ the union of infinite-dimensional ellipsoids embedded in $mathbbR^infty$ over $alphain A$?




    Thanks in advance!




    hilbert-spaces conic-sections




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      Let $X$ be a Hilbert manifold and $mathcalU_alpha$ an open subet of $X$, with a local coordinate chart $(mathcalU_alpha,phi_alpha)$ such that $X:=bigcup_alphain AmathcalU_alpha.$ Suppose the chart is given by $$phi_alpha:mathcalU_alphatomathbbR^infty \ gamma_alphamapsto (phi_alpha^1,phi_alpha^2,...).$$ Also suppose the local coordinates satisfy $$sum_k=1^inftyc_k^2||phi_alpha^k||_X^2=R_alpha^2$$ for $c_ine 0,i=1,...$.




      Then is $X$ the union of infinite-dimensional ellipsoids embedded in $mathbbR^infty$ over $alphain A$?




      Thanks in advance!




      hilbert-spaces conic-sections




      share|cite|improve this question













      Let $X$ be a Hilbert manifold and $mathcalU_alpha$ an open subet of $X$, with a local coordinate chart $(mathcalU_alpha,phi_alpha)$ such that $X:=bigcup_alphain AmathcalU_alpha.$ Suppose the chart is given by $$phi_alpha:mathcalU_alphatomathbbR^infty \ gamma_alphamapsto (phi_alpha^1,phi_alpha^2,...).$$ Also suppose the local coordinates satisfy $$sum_k=1^inftyc_k^2||phi_alpha^k||_X^2=R_alpha^2$$ for $c_ine 0,i=1,...$.




      Then is $X$ the union of infinite-dimensional ellipsoids embedded in $mathbbR^infty$ over $alphain A$?




      Thanks in advance!





      hilbert-spaces conic-sections





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Aug 4 at 5:10
























      asked Aug 3 at 3:27









      Multivariablecalculus

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