Differentiable vector-valued functions from Euclidean space
In the mathematical discipline of functional analysis, it is possible to generalize the notion of derivative to arbitrary (i.e. infinite dimensional) topological vector spaces (TVSs) in multiple ways. But when the domain of a TVS-value function is a subset of finite-dimensional Euclidean space then the number of generalizations of the derivative is much more limited and derivatives are more well behaved. This article presents the theory of -times continuously differentiable functions on an open subset of Euclidean space , which is an important special case of differentiation between arbitrary TVSs. All vector spaces will be assumed to be over the field where is either the real numbers or the complex numbers
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Differentiable vector-valued functions from Euclidean space
In the mathematical discipline of functional analysis, it is possible to generalize the notion of derivative to arbitrary (i.e. infinite dimensional) topological vector spaces (TVSs) in multiple ways. But when the domain of a TVS-value function is a subset of finite-dimensional Euclidean space then the number of generalizations of the derivative is much more limited and derivatives are more well behaved. This article presents the theory of -times continuously differentiable functions on an open subset of Euclidean space , which is an important special case of differentiation between arbitrary TVSs. All vector spaces will be assumed to be over the field where is either the real numbers or the complex numbers
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In the mathematical discipline ...... numbers or the complex numbers
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63,944,266
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If is a complete Hausdorff lo ...... o the injective tensor product
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Let be a Hausdorff locally co ...... ction
:
is also continuous .
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Let be a positive integer and ...... :
is an isomorphism of TVSs .
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Theorem
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In the mathematical discipline ...... numbers or the complex numbers
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Differentiable vector-valued functions from Euclidean space
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