What is a countably infinite-dimensional coordinate space called and where can I read more about it?

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The countably-infinite-dimensional generalization of Euclidean space is $l^{2}$, and is a Hilbert Space. This space is named after the Mathematician who initiated the first serious study of such a space around 1905. The sequences $(x_{1},x_{2},x_{3},\ldots)$ are assumed to satisfy $\sum_{n=1}^{\infty}|x_{n}|^{2} < \infty$.

http://en.wikipedia.org/wiki/Hilbert_space

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evencoil
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evencoil

Updated on August 01, 2022

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  • evencoil
    evencoil over 1 year

    This is just a question about terminology. I'm interested in spaces of vectors with countably many components $(x_{1},x_{2},x_{3},\ldots)$ where each $x_{k} \in \mathbb{R}$. What are these spaces called? On Wikipedia (http://en.wikipedia.org/wiki/Examples_of_vector_spaces#Infinite_coordinate_space) they are referred to just as "Infinite coordinate space" but this term hasn't been very helpful when put into a Google search.

    More specifically, I want to see what is known about linear algebra type problems when the vector lengths are countably infinite. Is there a name for this topic? Can you recommend good introductory books?

    • Dave L. Renfro
      Dave L. Renfro almost 10 years
      The wikipedia article titled sequence space seems to be a better fit for what you want. In class these spaces are often called "little elle-p spaces", but for googling purposes this looks like it would be a good start: l^p space sequences.