# Space of Bounded Sequences with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space

## Theorem

Let $\map {\ell^\infty} \C$ be the space of bounded sequences on $\C$.

Let $\struct {\C, +_\C, \times_\C}$ be the field of complex numbers.

Let $\paren +$ be the pointwise addition on the ring of sequences.

Let $\paren {\, \cdot \,}$ be the pointwise multiplication on the ring of sequences.

Then $\struct {\map {\ell^\infty} \C, +, \, \cdot \,}_\C$ is a vector space.

## Proof

Let $\sequence {a_n}_{n \mathop \in \N}, \sequence {b_n}_{n \mathop \in \N}, \sequence {c_n}_{n \mathop \in \N} \in \map {\ell^\infty} \C$.

Let $\lambda, \mu \in \C$.

Let $\sequence 0 := \tuple {0, 0, 0, \dots}$ be a complex-valued function.

Let us use real number addition and multiplication.

$\sequence {a_n + b_n}_{n \mathop \in \N} := \sequence {a_n}_{n \mathop \in \N} +_\C \sequence {b_n}_{n \mathop \in \N}$.

Define pointwise scalar multiplication as:

$\sequence {\lambda \cdot a_n}_{n \mathop \in \N} := \lambda \times_\C \sequence {a_n}_{n \mathop \in \N}$

Let the additive inverse be $\sequence {-a_n} := - \sequence {a_n}$.

### Closure Axiom

By assumption, $\sequence {a_n}_{n \mathop \in \N}, \sequence {b_n}_{n \mathop \in \N} \in \map {\ell^\infty} \C$.

By definition:

$\ds \sup_{n \mathop \in \N} \size {a_n} < \infty$
$\ds \sup_{n \mathop \in \N} \size {b_n} < \infty$

Consider the sequence $\sequence {a_n + b_n}$.

Then:

 $\ds \sup_{n \mathop \in \N} \size {a_n + b_n}$ $\le$ $\ds \sup_{n \mathop \in \N} \size {\size {a_n} + \size {b_n} }$ $\ds$ $\le$ $\ds \sup_{n \mathop \in \N} \size {\map \max {\size {a_n}, \size {b_n} } + \map \max {\size {a_n}, \size {b_n} } }$ Definition of Max Operation $\ds$ $=$ $\ds \sup_{n \mathop \in \N} \size {2 \map \max {\size {a_n}, \size {b_n} } }$ $\ds$ $\le$ $\ds 2 \paren {\sup_{n \mathop \in \N} \size {a_n} + \sup_{n \mathop \in \N} \size {b_n} }$ $\ds$ $<$ $\ds \infty$ $\sequence {a_n}, \sequence {b_n} \in \map {\ell^\infty} \C$

Hence:

$\sequence {a_n + b_n} \in \map {\ell^\infty} \C$

$\Box$

### Commutativity Axiom

By Pointwise Addition on Ring of Sequences is Commutative, $\sequence {a_n} + \sequence {b_n} = \sequence {b_n} + \sequence {a_n}$

$\Box$

### Associativity Axiom

By Pointwise Addition on Ring of Sequences is Associative, $\paren {\sequence {a_n} + \sequence {b_n} } + \sequence {c_n} = \sequence {a_n} + \paren {\sequence {b_n} + \sequence {c_n} }$.

$\Box$

### Identity Axiom

 $\ds \sequence {0 + a_n}$ $=$ $\ds \sequence 0 +_\C \sequence {a_n}$ Definition of Pointwise Addition on Ring of Sequences $\ds$ $=$ $\ds \tuple {0, 0, 0, \dots} +_\C \sequence {a_n}$ Definition of $\sequence 0$ $\ds$ $=$ $\ds \sequence {a_n}$

$\Box$

### Inverse Axiom

 $\ds \sequence {a_n + \paren {-a_n} }$ $=$ $\ds \sequence {a_n} +_\C \sequence {-a_n}$ Definition of Pointwise Addition on Ring of Sequences $\ds$ $=$ $\ds \sequence {a_n} +_\C \paren {-1} \times_\C \sequence {a_n}$ Definition of $\sequence {-a_n}$ $\ds$ $=$ $\ds 0$

$\Box$

 $\ds \sequence {\paren {\lambda +_\C \mu} \cdot a_n }$ $=$ $\ds \paren {\lambda +_\C \mu} \times_\C \sequence {a_n}$ Definition of Pointwise Scalar Multiplication on Ring of Sequences $\ds$ $=$ $\ds \lambda \times_\C \sequence {a_n} +_\C \mu \times_\C \sequence {a_n}$ Real Multiplication Distributes over Addition $\ds$ $=$ $\ds \sequence {\lambda \cdot a_n} +_\C \sequence {\mu \cdot a_n}$ Definition of Pointwise Scalar Multiplication on Ring of Sequences $\ds$ $=$ $\ds \sequence {\lambda \cdot a_n + \mu \cdot a_n}$ Definition of Pointwise Addition on Ring of Sequences

$\Box$

 $\ds \lambda \times_\C \sequence {a_n + b_n}$ $=$ $\ds \lambda \times_\C \paren {\sequence {a_n} +_\C \sequence {b_n} }$ Definition of Pointwise Addition on Ring of Sequences $\ds$ $=$ $\ds \lambda \times_\C \sequence {a_n} +_\C \lambda \times_\C \sequence {b_n}$ Real Multiplication Distributes over Addition $\ds$ $=$ $\ds \sequence {\lambda \cdot a_n} +_\C \sequence {\lambda \cdot b_n}$ Definition of Pointwise Scalar Multiplication on Ring of Sequences $\ds$ $=$ $\ds \sequence {\lambda \cdot a_n + \mu \cdot b_n}$ Definition of Pointwise Addition on Ring of Sequences

$\Box$

### Associativity with Scalar Multiplication

 $\ds \sequence {\paren {\lambda \times_\C \mu} \cdot a_n}$ $=$ $\ds \paren {\lambda \times_\C \mu} \times_\C \sequence {a_n}$ Definition of Pointwise Scalar Multiplication on Ring of Sequences $\ds$ $=$ $\ds \lambda \times_\C \paren {\mu \times_\C \sequence {a_n} }$ Real Multiplication is Associative $\ds$ $=$ $\ds \lambda \times_\C \sequence {\mu \cdot a_n}$ Definition of Pointwise Scalar Multiplication on Ring of Sequences $\ds$ $=$ $\ds \sequence {\lambda \cdot \paren {\mu \cdot a_n} }$ Definition of Pointwise Scalar Multiplication on Ring of Sequences

$\Box$

### Identity for Scalar Multiplication

 $\ds \sequence {1 \cdot a_n}$ $=$ $\ds 1 \times_\C \sequence {a_n}$ Definition of Pointwise Scalar Multiplication on Ring of Sequences $\ds$ $=$ $\ds \sequence {a_n}$

$\Box$

$\blacksquare$