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

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Theorem

Let $\ell^\infty$ be the space of bounded sequences.

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


Proof

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

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.

Define pointwise addition as:

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

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

By definition:

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

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

Then:

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

Hence:

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

$\Box$


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

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

$\Box$


Inverse Axiom

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

$\Box$


Distributivity over Scalar Addition

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

$\Box$


Distributivity over Vector Addition

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

$\Box$


Associativity with Scalar Multiplication

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

$\Box$


Identity for Scalar Multiplication

\(\displaystyle \sequence {1 \cdot a_n}\) \(=\) \(\displaystyle 1 \times_\C \sequence {a_n}\) Definition of Pointwise Scalar Multiplication on Ring of Sequences
\(\displaystyle \) \(=\) \(\displaystyle \sequence {a_n}\)

$\Box$

$\blacksquare$


Sources