grandes-ecoles 2020 Q36

grandes-ecoles · France · centrale-maths2__mp Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions

Let $(E, \langle \cdot, \cdot \rangle)$ be a reproducing kernel Hilbert space on an interval $I$, with reproducing kernel $K$. For all $(x,y) \in I^2$, we set $k_x(y) = K(x,y)$. Suppose that $K$ is continuous on $I \times I$. Prove that all functions in $E$ are continuous.

Let $(E, \langle \cdot, \cdot \rangle)$ be a reproducing kernel Hilbert space on an interval $I$, with reproducing kernel $K$. For all $(x,y) \in I^2$, we set $k_x(y) = K(x,y)$.
Suppose that $K$ is continuous on $I \times I$.
Prove that all functions in $E$ are continuous.

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