关于n维整数空间的存在性及其与{𝐱∈ℝⁿ|x₁,…,xₙ∈ℤ}等价性的问询

Great question—let’s break this down clearly and directly:

Yes, the n-dimensional integer space absolutely exists, and it is exactly equivalent to the set {𝐱∈ℝⁿ | x₁, x₂, …, xₙ ∈ ℤ} you described.

Here’s a deeper breakdown to confirm your intuition:

• Standard Definition & Notation: In mathematics, this space is universally denoted as ℤⁿ. By definition, ℤⁿ is the collection of all n-tuples where every component is an integer. This is precisely the same as your proposed set: every element is a vector in ℝⁿ with integer coordinates. The two are entirely identical.
• Why It’s Called a "Space": While ℤⁿ isn’t a Euclidean vector space (it doesn’t close under scalar multiplication by real numbers—for example, (1,0) ∈ ℤ² multiplied by 0.5 gives (0.5,0), which isn’t in ℤ²), it’s still a robust, well-studied algebraic structure. It’s a free abelian group under component-wise addition, and forms a lattice within ℝⁿ—a discrete subgroup that spans the ambient real vector space. These properties earn it the label "space" in broader algebraic and number-theoretic contexts.
• Real-World Use Cases: ℤⁿ is foundational across multiple fields:
  • Number theory (analyzing Diophantine equations, which require integer solutions)
  • Combinatorics (counting discrete configurations and arrangements)
  • Computer science (discrete optimization problems, lattice-based cryptography)

To sum up: Your thinking is exactly right—the n-dimensional integer space is precisely the set of integer-coordinate vectors in ℝⁿ, and it’s a core, widely used mathematical object.

内容的提问来源于stack exchange，提问作者The Pointer