P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences form Vector Space

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Theorem

Let $\ell^p$ be the p-sequence space.

Let $\struct {\R, +_\R, \times_\R}$ be the field of real 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 $\struct {\ell^p, +, \, \cdot \,}_\R$ is a vector space.


Proof

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

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

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

Let us use real number addition and multiplication.

Define pointwise addition as:

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

Define pointwise scalar multiplication as:

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

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


Closure Axiom

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

By definition:

$\displaystyle \sum_{n \mathop = 1}^\infty \size {a_n}^p < \infty$
$\displaystyle \sum_{n \mathop = 1}^\infty \size {b_n}^p < \infty$

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

Then:

\(\displaystyle \sum_{n \mathop = 1}^\infty \size {a_n + b_n}^p\) \(\le\) \(\displaystyle \sum_{n \mathop = 1}^\infty \paren {\size {a_n} + \size {b_n} }^p\)
\(\displaystyle \) \(\le\) \(\displaystyle \sum_{n \mathop = 1}^\infty \paren {\map \max {\size {a_n}, \size {b_n} } + \map \max {\size {a_n}, \size {b_n} } }^p\) Definition of Max Operation
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop = 1}^\infty 2^p \paren {\map \max {\size {a_n}, \size {b_n} } }^p\)
\(\displaystyle \) \(\le\) \(\displaystyle 2^p \sum_{n \mathop = 1}^\infty \paren {\size {a_n}^p + \size {b_n}^p}\)
\(\displaystyle \) \(<\) \(\displaystyle \infty\) $\sequence {a_n}, \sequence {b_n} \in \ell^p$

Hence:

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

$\Box$


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

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

$\Box$


Inverse Axiom

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

$\Box$


Distributivity over Scalar Addition

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

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

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

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

$\Box$

$\blacksquare$


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