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On a Riemannian manifold $(M,g)$, let $\mathcal L^p(M,k)$ denote the space of measurable $k$-tensors $T$ (i.e., the coordinate components in any chart are Lebesgue measurable) for which the norm $$||T||_p=\int_M |T|^p\,d\mu_g,\quad |T|=\sqrt{T_{i_1\cdots i_k}T^{i_1\cdots i_k} }$$ is finite. Is this space complete?

In compact case, I obtained an affirmative answer on Math Stack ExchangeMath Stack Exchange, but the answer does not seem to carry over to the general case. The book by Chavel, Eigenvalues in Riemannian Geometry, seems to claim this is true. He offers no proof.

On a Riemannian manifold $(M,g)$, let $\mathcal L^p(M,k)$ denote the space of measurable $k$-tensors $T$ (i.e., the coordinate components in any chart are Lebesgue measurable) for which the norm $$||T||_p=\int_M |T|^p\,d\mu_g,\quad |T|=\sqrt{T_{i_1\cdots i_k}T^{i_1\cdots i_k} }$$ is finite. Is this space complete?

In compact case, I obtained an affirmative answer on Math Stack Exchange, but the answer does not seem to carry over to the general case. The book by Chavel, Eigenvalues in Riemannian Geometry, seems to claim this is true. He offers no proof.

On a Riemannian manifold $(M,g)$, let $\mathcal L^p(M,k)$ denote the space of measurable $k$-tensors $T$ (i.e., the coordinate components in any chart are Lebesgue measurable) for which the norm $$||T||_p=\int_M |T|^p\,d\mu_g,\quad |T|=\sqrt{T_{i_1\cdots i_k}T^{i_1\cdots i_k} }$$ is finite. Is this space complete?

In compact case, I obtained an affirmative answer on Math Stack Exchange, but the answer does not seem to carry over to the general case. The book by Chavel, Eigenvalues in Riemannian Geometry, seems to claim this is true. He offers no proof.

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Ryan Unger
Ryan Unger
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Is the $L^p$ space of tensors complete?

On a Riemannian manifold $(M,g)$, let $\mathcal L^p(M,k)$ denote the space of measurable $k$-tensors $T$ (i.e., the coordinate components in any chart are Lebesgue measurable) for which the norm $$||T||_p=\int_M |T|^p\,d\mu_g,\quad |T|=\sqrt{T_{i_1\cdots i_k}T^{i_1\cdots i_k} }$$ is finite. Is this space complete?

In compact case, I obtained an affirmative answer on Math Stack Exchange, but the answer does not seem to carry over to the general case. The book by Chavel, Eigenvalues in Riemannian Geometry, seems to claim this is true. He offers no proof.