# Questions tagged [holder-inequality]

Proving or manipulations with inequalities by using Hölder's inequality.

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### For $1<p<q<\infty$ show $L^q[0, 1] \subset L^p[0, 1]$ [closed]

For $1<p<q<\infty$ show $L^q[0, 1] \subset L^p[0, 1]$ I know I'm supposed to use holders inequality to solve this, but can't work it out. We haven't learnt anything about measure theory, so ...
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### Vectors such that $\lVert x + y \rVert_p^p = \lVert x \rVert_p^p + \lVert y \rVert_p^p$

We fix $1 \leq p < 2$. What are the couple $({x},{y})$ of vectors in $\mathbb{R}^2$ (or more generally in $\mathbb{R}^n$) for which the following equality holds \begin{equation} \label{eq:here} \...
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### Hölder inequality, showing that if $\int_1^\infty |f(x)| dx$ exists, then exists $\int_1^\infty |f(x)|^2 dx$

So i have a Problem with a proof. given is a function $f:[1,\infty) \to R$, which is in every intervall $[1,b]$ with $b>1$ Riemann-Integratable. There are also given $p,q,r ∈ R$ with $1≤p≤r≤q$. ...
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### Application of Arzelà-Ascoli theorem

I am trying to understand the wikipedia article on the Arzelà-Ascoli theorem, which can be found here: https://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem More specifically, I am trying ...
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