What is the correct notation to define a vector in $\mathbb{R}^n$ within the interval $[0,1]$?

Question

If I can can define a binary string with $n$ bits as a vector in the space $\{0,1\}^n$; how can I define a vector in $\mathbb{R}^n$ for the interval $[0,1]$? Can I just write $[0,1]^n$?

Answer 1

Let $A$ be an arbitrary set. Then $A^n$ denotes the set $A\times A\times \dots \times A$, where $A$ appears $n$ times in the cartesian product. You can call $A^n$ a cartesian power.

Therefore $A^n$ is the set of $n$-tuples $(a_1,a_2,\dots,a_n)$ where each $a_i$ belongs to $A$.

This applies to any set $A$. So, $\Bbb R^n$ is the set of $n$-tuples of reals, and $[0,1]^n$ is the set of $n$-tuples of numbers between $0$ and $1$. And $\{0,1\}^n$ is the set of bit strings of length $n$.

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Side note: to consider $\{0,1\}^n$ a vector space, you have first to define a structure of field on $\{0,1\}$. See GF(2). And $[0,1]^n$ is not a vector space, though it's a subset of the vector space $\Bbb R^n$.