What is the cardinality of $\mathbb{R^n}$ and $l^2$?

Asked Sep 03 '21 at 09:06

Active Sep 03 '21 at 10:33

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$\mathfrak{c}$ is cardinaity of real numbers. It is also called the cardinality of continuum. It makes sense to me that this cardinality is strictly bigger than cardinality of $\mathbb{N}$ and $\mathbb{Q}$. But what about relations to other cardinalities?

What is the cardinality of $\mathbb{R^n}$? Is it the same, because it is just "higher-dimensional $\mathbb{R}$, or is it $n$ times the cardinality of $\mathbb{R}$?
What is the cardinality of $l^2$ which is like "$\mathbb{R^{\infty}}$"?

Thank you for providing an answer and potentially a general approach to how to determine cardinalities of sets.

edited Sep 03 '21 at 10:33

asked Sep 03 '21 at 09:06

Tereza Tizkova

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Do you know what it means to say two sets have the same cardinality? – Infinity_hunter Sep 03 '21 at 09:15
After you learn what cardinality is (It is very basic i think) check Cantor-Schröder–Bernstein theorem – Oğuzhan Kılıç Sep 03 '21 at 09:19
@Infinity_hunter I would say they have the same "size", but it doesn´t tell me so much about the sets. – Tereza Tizkova Sep 03 '21 at 09:48
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@TerezaTizkova I suggest you look here to see the definition. – Infinity_hunter Sep 03 '21 at 09:53
@SaucyO'Path I edited, you are right, my formulation was not very bright. – Tereza Tizkova Sep 03 '21 at 10:35
Note that $|\mathbb R^n| = |\mathbb R|^n = 2^{\aleph_0 \cdot n} = 2^{\aleph_0} = |\mathbb R|$. Also note that by choice for infinite $X,Y$, $|X \times Y| = \operatorname{max}(|X|,|Y|)$. – Tan Sep 03 '21 at 10:35

What is the cardinality of $\mathbb{R^n}$ and $l^2$?

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