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Questions tagged [lp-spaces]

For questions about $L^p$ spaces, that is, given a measure space $(X,\mathcal F,\mu)$, the vector space of equivalence class of measurable functions with $p$-th power of the absolute value integrable. Question can be about properties of elements of these spaces, or when the ambient space on a problem is a $L^p$ space.

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Is it true that a sequence $f_n \to f$ of measurable functions is bounded by a norm of $L^p$ then $f_n$ converges to $f$ in $L^p$? [duplicate]

Is it true that a sequence $f_n \to f$ of measurable functions is bounded by a norm of $L^p$ then $f_n$ converges to f in $L^p$? Is this true? If so, prove, if not, a counter example. I just ...
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Splitting function from $L^{n/2}$

Let $V\in L^{n/2}$, $n\geq 3$. I want to show that for every $\varepsilon>0$, there are $||V_{1}||_{L^{n/2}}\leq \varepsilon$ and $V_{2}\in L^{\infty}$ sucht that $$V=V_{1}+V_{2}$$
Let X be a measurable set with $\mu (X) < \infty$ and $1 \leq p < \infty$… ($L^p$ spaces
Let X be a measurable set with $\mu (X) < \infty$ and $1 \leq p < \infty$ Let $(f_n)_{n \in \mathbb{N}} \subset X$ and $f \in L^{p}(X)$ with $\lim_{n \to \infty}||f_n - f|| = 0$ Show ...