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

Either L or ℓ denotes the first alphabetic letter of Lebesgue. According to the definitions, there are interpretations for both L^p spaces and ℓ^p space. Please note that Frigyes Riesz named L^p spaces with adding the pulral "s" to the word "space". Mr. Riesz proved the L^2 spaces and ℓ^2 space are isomorphic. Of course, ℓ^2 space is the Hilbert space.
1.L^p (function) spaces
L^p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. L^p spaces is called Lebesgue spaces, named after Henri Lebesgue.
2.ℓ^p (sequence) space
The above-mentioned p-norm can be extended to vectors that have an infinite number of components (sequences), which yields ℓ^p space. ℓ^p space is the sequence space consisting of the p-power summable sequences with the p-norm. In addition, it has bounded sequences.
However, some mathematicians used both L^p spaces and ℓ^p spaces to mention the Banach space, and both L^2 spaces and ℓ^2 space for the Hilbert space interchangeably (because infinite-dimensional vectors are used in L^p function spaces sometimes even though the usage is not strict in the math). The mathematicians have avoided the subtle difference of the two concepts. It makes people confused sometimes.
