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Given a connection on the tangent bundle, you can define the second covariant derivative $\nabla^2f$ by $$ \nabla^2f[X, Y] := \partial_X\partial_Y f - \partial_{\nabla_X Y} f.$$ Then $\nabla^2f$ is a …

It is easy to see that the dual space of a bornological space $V$ (i.e. the space of bounded linear functionals) may be zero (just take any vector space with the power set as bornology). Hence in gene …

For a compact topological space $X$, denote by $\mathcal{M}(X)$ the Banach space of finite signed Borel (Radon) measures on $X$ with the total variation norm. This is canonically isometric to the dual …

Let $A$ be an algebra over $\mathbb{C}$. Let $M$ be a left $A$-module, let $N$ be a right $A$-module and consider the tensor product $N \otimes_A M$, which is a complex vector space. Q1: Can this ten …

For a locally convex Hausdorff spaces $E$, consider the canonical map $$\overline{\psi}:E^\prime \hat{\otimes}_\pi E \longrightarrow L(E_\sigma)$$ that maps the projective tensor product to the space …