Is $L_p, loc(R)$ a Banach algebra?

Asked May 04 '18 at 15:11

Active May 04 '18 at 15:11

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I am not an expert in functional analysis, and I would like to know whether the space of locally integrabile functions, $L_{1,loc}(R)$ is a Banach algebra with respect to a suitable norm.

The Wikipedia entry https://en.m.wikipedia.org/wiki/Locally_integrable_function

discusses a possible metric, but leaves my question open. I’d be happy to be forwarded to a relatively accessible reference, if available.

asked May 04 '18 at 15:11

Guran Semiotovic

What muitplication do you have in mind? – David C. Ullrich May 04 '18 at 15:31
Thanks @DavidC.Ullrich. I was thinking point wise multiplication between functions, $(uv)(x) = u(x)v(x)$. – Guran Semiotovic May 04 '18 at 22:23
But the pointwise product of two locally integrable functions need not be locally integrable. – David C. Ullrich May 04 '18 at 22:30
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(When people talk about $L^1(\Bbb R)$ as a Banach algebra the multiplication is convolution. But that doesn't work here either...) – David C. Ullrich May 04 '18 at 22:35
I see, thank you. That answers my question, thanks – Guran Semiotovic May 05 '18 at 05:13
Hi David, do you know an example of a product and norm under which $L_{1,loc}(\mathbb{R})$ is a Banach Algebra? – Guran Semiotovic Jun 16 '18 at 21:35

Is $L_p, loc(R)$ a Banach algebra?

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