Banach Spaces

Banach Spaces - Example #1

Given:

• ℝ is a one-dimensional real vector space
• ||·|| : ℝ → [0, ∞] is a norm

Thus:

• 𝑑_||·||(𝑥,𝑦) = |𝑥-𝑦| is a distance metric
• (ℝ,𝑑_||·||) is a Banach space

Banach Spaces - Example #2

Given:

• 𝑉 is a zero-dimensional real vector space
• ||·|| : 𝑉 → [0, ∞] is a norm defined by ||0|| = 0

Thus:

• (𝑉,||·||) is a Banach space

Banach Spaces - Example #3

Given:

• ℕ is the set of natural numbers
• 𝔽 is a field of real and/or complex numbers
• 𝑝 ∊ [1,∞)

Let 𝐿^𝑝(ℕ,𝔽) an Lp space be defined as all sequences (𝑥_𝑛)_𝑛∊ℕ in 𝔽 such that:

• ${\sum }_{n=1}^{\infty }|x_n{|}^p<\infty (converges!)$

Then ||·||_𝑝 : 𝐿^𝑝 → [0, ∞) is the norm defined as:

• $||x|{|}_p=({\sum }_{n=1}^{\infty }|x_n{|}^p{)}^{1/p}$

(𝐿^𝑝,||·||_𝑝) is a Banach space.