Vector valued functions equivalent to measurable functions

On Spaces of Measurable Vector - Valued Functions 399 Proof

Holomorphic vector-valued functions

exists. The function f is continuously differentiable when it is differentiable and f ′ is continuous. A k-times continuously differentiable function is C, and a continuous function is C. A V -valued function f is weakly C when for every λ ∈ V ∗ the scalar-valued function λ◦ f is C. This sense of weak differentiability of a function f does not refer to distributional derivatives, but to differe...

Differentiation of Vector-Valued Functions

This important result is listed in the theorem on the next page. Note that the derivative of the vector-valued function is itself a vector-valued function. You can see from Figure 12.8 that is a vector tangent to the curve given by and pointing in the direction of increasing values. tr t r t r f t i g t j lim t→0 f t t f t t i lim t→0 g t t g t t j lim t→0 f t t f t t i g t t g t t j lim t→0 f ...

On Learning Vector-Valued Functions

In this letter, we provide a study of learning in a Hilbert space of vectorvalued functions. We motivate the need for extending learning theory of scalar-valued functions by practical considerations and establish some basic results for learning vector-valued functions that should prove useful in applications. Specifically, we allow an output space Y to be a Hilbert space, and we consider a repr...

Generalized Rings of Measurable and Continuous Functions

This paper is an attempt to generalize, simultaneously, the ring of real-valued continuous functions and the ring of real-valued measurable functions.