Banach Spaces

• is a type of mathematical space
• is a vector space (𝑉,𝐹) over the real or complex numbers on which a norm (||·||) is defined and is complete
• is a normed vector space (𝑉,𝐹,||·||) with completeness
• is both a complete metric space and a real/complex vector space tied together by the norm
• the chosen norm (||·||) implicitly defines a distance metric (𝑑_||·||); thus making a Banach space a special case of a metric space

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:

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

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(𝐿^𝑝,||·||_𝑝) is a Banach space.