Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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Continuity of Function Related to $F$-norms

Let $X$ be a locally bounded $F$-space and $\left\|\cdot\right\|$ be an $F$-norm on $X$. Suppose that $\left\|\cdot\right\|$ is concave: for all $x\in X$ fixed, the function ...
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weak derivative questions

There are two definitions of generalized differentiation that seem relevant to the context of PDEs. (That is we generalize what objects can be differentiated but we stay in Euclidean space. There are ...
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Mean of Piecewise function resting on IID random variables

Suppose IID random variables $X_t \sim X$ with support on $[0,1]$ and continuous CDF $F(\cdot)$. I wish to compute the expected value (mean) of the a piecewise function with form $$ \Phi (x,\mu) = ...