Differences between $L^p$ and $\ell^p$ spaces

Could someone explain some differences between the $L^p$ and $\ell^p$ spaces? Thanks a lot.

Solution 1:

Typically, $\ell^p$ is used to indicate a $p$-summable discrete set of values. For example, $\ell^p(\mathbb{Z}^{+})$ is the set of complex-valued sequences $\{(a_n)\}$ such that $\sum_{n \in \mathbb{Z}^{+}} |a_n|^p < \infty$.

$L^p$ is typically used to indicate $p$-summable functions (with respect to some measure) on a non-discrete measure space, such as the usual $L^p(\mathbb{R})$, the set of functions $f: \mathbb{R} \rightarrow \mathbb{C}$ such that $\int_{\mathbb{R}} |f(x)|^p \, dx < \infty$.

The main point is that they are mathematically different notation for the same concept. If we think of $\mathbb{Z}^{+}$ as a discrete measure space and take the uniform measure on it, then a complex-valued sequence $(a_n)$ is a complex-valued measurable function on $\mathbb{Z}^{+}$.

Solution 2:

Of course, Christopher's answer is correct ($\ell^p$ is just a particular instance of a Lebesgue space $L^p$), but there are two other implicit questions here: how are the spaces $\ell^p$ and $L^p$ different. and what role could $p$ play?

This depends on what exactly you mean by $L^p$. Perhaps you could be thinking of either $L^p([0,1])$, or $L^p(\mathbb{R})$. For a fixed choice of $p$, these spaces aren't all that different, but if you consider the scale of $L^p$ spaces (that is, if you consider all $p \in (0,\infty)$), you start to see some differences. In particular, if $1 \leq p < q \leq \infty$, then $L^q([0,1])$ is contained in $L^p([0,1])$, by Hölder's inequality (so the $L^p([0,1])$ scale is decreasing). For $p=1$ and $q=\infty$, this is just the fact that a bounded function on $[0,1]$ is integrable. However, there are no containment relations between the spaces $L^p(\mathbb{R})$. In this sense, $L^p([0,1])$ and $L^p(\mathbb{R})$ are different.

On the other hand, thinking now about the spaces $\ell^p(\mathbb{N})$, the situation is a bit like $L^p([0,1])$ but reversed - if $1 \leq p < q < \infty$, then $\ell^p(\mathbb{N})$ is contained in $\ell^q(\mathbb{N})$, and so the $\ell^p(\mathbb{N})$ scale is increasing. For $p=1$ and $q=\infty$, this is the fact that a summable sequence is bounded. In this sense, $\ell^p(\mathbb{N})$ is different to both $L^p([0,1])$ and $L^p(\mathbb{R})$!

More generally, what I've said remains true if I replace $[0,1]$ with a measure space of finite measure and $\mathbb{N}$ with a granular measure space - that is, a measure space such that the measure of any nonempty measurable set is bounded below by some positive constant.

I'm sure there are some more subtle differences between $\ell^p$ and $L^p$ as Banach spaces (see for example Stephen Montgomery-Smith's comment) but I don't know much about this.