Introduction

Definition 1: Vector space

A linear vector space S S is a collection of "vectors" such that (1) if f 1 ∈S⇒α f 1 ∈S f 1 S α f 1 S for all scalars αα (where α∈ℝ α or α∈ℂ α ) and (2) if f 1 ∈S f 1 S , f 2 ∈S f 2 S , then f 1 + f 2 ∈S f 1 f 2 S

Vector Spaces

If the scalars αα are real, SS is called a real vector space.

If the scalars αα are complex, SS is called a complex vector space.

If the "vectors" in SS are functions of a continuous variable, we sometimes call SS a linear function space

Examples

• ℝn=real vector space n real vector space
• ℂn=complex vector space n complex vector space
• L 1 ℝ={ft|∫-∞∞|ft|dt<∞} L 1 f t t f t f t is a vector space
• L ∞ ℝ={ft| f ⁡ ( t )  is bounded } L ∞ f t f ⁡ ( t )  is bounded f t is a vector space
• L 2 ℝ={ft|∫-∞∞|ft|2dt<∞}=finite energy signals L 2 f t t f t 2 f t finite energy signals is a vector space
• L 2 0T =finite energy functions on interval [0,T] L 2 0 T finite energy functions on interval [0,T]
• ℓ 1 ℤ ℓ 1 , ℓ 2 ℤ ℓ 2 , ℓ ∞ ℤ ℓ ∞ are vector spaces
• The collection of functions piecewise constant between the integers is a vector space

Figure 1

• ℝ + 2={ x 0 x 1 | x 0 >0∧ x 1 >0} ℝ + 2 x 0 x 1 x 0 0 x 1 0 x 0 x 1 is not a vector space. 11∈ ℝ + 2 1 1 ℝ + 2 , but ∀α,α<0:α11∉ ℝ + 2 α α 0 α 1 1 ℝ + 2
• D=∀z,|z|≤1:z∈ℂ D z z 1 z is not a vector space. z 1 =1∈D z 1 1 D , z 2 =ⅈ∈D z 2 D , but z 1 + z 2 ∉D z 1 z 2 D , | z 1 + z 2 |=2>1 z 1 z 2 2 1

note:

Vector spaces can be collections of functions, collections of sequences, as well as collections of traditional vectors (i.e. finite lists of numbers)