# Tagged Questions

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### Prove that $L^1$ is a Banach algebra with multiplication defined by convolution

To be more specific, prove that $L^1(\mathbb{R}^n)$ with multiplication defined by convolution: $$(f\cdot g)(x)=\int_\mathbb{R^n}f(x-y)g(y)dy$$ is a Banach algebra. All the properties of Banach ...
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### Do inequalities that hold for infinite sums hold for integrals too?

Let $\mathbb{R}_{\geq0}$ denote the set of non-negative reals and $+\infty$, and $\mathbb{Z}^+$ denote the set of positive integers. I will also let $\lambda$ denote the Lebesgue measure on ...
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### Hölder inequality from Jensen inequality

I'm taking a course in Analysis in which the following exercise was given. Exercise Let $(\Omega, \mathcal{F}, \mu)$ be a probability space. Let $f\ge 0$ be a measurable function. Using Jensen's ...
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