Infinite-Dimensional Vector Spaces (𝑉,𝐹)

• is a vector space (𝑉,𝐹) that can NOT be spanned by a FINITE set of basis vectors
• opposed to finite-dimensional vector space

Infinite-Dimensional Vector Space - Basis

see: Hamel Basis Vectors - Basis for Infinite-Dimensional Vector Spaces

Infinite-Dimensional Vector Space - Examples

• the vector space ℙ(ℝ) of all polynomial functions on the real line
• the set of all polynomial functions of ANY degree
  1. example infinite basis {1, 𝑥, 𝑥², ... } # "standard basis"
  2. example infinite basis {4, 6𝑥, 1𝑥², ... }
• the set of all functions 𝑓: ℝ → ℝ
  1. example infinite basis #1 {1, 𝑥, 𝑥², ..., 𝑐𝑜𝑠(𝑥), ..., 5^𝑥, ... }
• the set of all continuous functions 𝑓: ℝ → ℝ
  • the set of all continuous functions 𝑓: ℝ → ℝ on the interval [0, 1]
• the set of all differentiable functions from ℝ to itself. Generally, we can talk about other families of functions that are closed under addition and scalar multiplication
• vector space of all sequences (𝐹^ℕ)
  • the set of all infinite sequences over ℝ
  • the set of all infinite sequences over ℂ
• Any 𝐿^𝑝 spaces
  
• The Hilbert Cube I∞
  
• The Tikhonov cube Iτ

Other

• Vector Spaces - Finite vs Infinite Dimensions