P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms 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:
- $\ds \sum_{n \mathop = 1}^\infty \size {a_n}^p < \infty$
- $\ds \sum_{n \mathop = 1}^\infty \size {b_n}^p < \infty$
Consider the sequence $\sequence {a_n + b_n}$.
Then:
\(\ds \sum_{n \mathop = 1}^\infty \size {a_n + b_n}^p\) | \(\le\) | \(\ds \sum_{n \mathop = 1}^\infty \paren {\size {a_n} + \size {b_n} }^p\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \sum_{n \mathop = 1}^\infty \paren {\map \max {\size {a_n}, \size {b_n} } + \map \max {\size {a_n}, \size {b_n} } }^p\) | Definition of Max Operation | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty 2^p \paren {\map \max {\size {a_n}, \size {b_n} } }^p\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds 2^p \sum_{n \mathop = 1}^\infty \paren {\size {a_n}^p + \size {b_n}^p}\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \infty\) | $\sequence {a_n}, \sequence {b_n} \in \ell^p$ |
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
\(\ds \sequence {0 + a_n}\) | \(=\) | \(\ds \sequence 0 +_\R \sequence {a_n}\) | Definition of Pointwise Addition on Ring of Sequences | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {0, 0, 0, \dots} +_\R \sequence {a_n}\) | Definition of $\sequence 0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sequence {a_n}\) |
$\Box$
Inverse Axiom
\(\ds \sequence {a_n + \paren {-a_n} }\) | \(=\) | \(\ds \sequence {a_n} +_\R \sequence {-a_n}\) | Definition of Pointwise Addition on Ring of Sequences | |||||||||||
\(\ds \) | \(=\) | \(\ds \sequence {a_n} +_\R \paren {-1} \times_\R \sequence {a_n}\) | Definition of $\sequence {-a_n}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
$\Box$
Distributivity over Scalar Addition
\(\ds \sequence {\paren {\lambda +_\R \mu} \cdot a_n }\) | \(=\) | \(\ds \paren {\lambda +_\R \mu} \times_\R \sequence {a_n}\) | Definition of Pointwise Scalar Multiplication on Ring of Sequences | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \times_\R \sequence {a_n} +_\R \mu \times_\R \sequence {a_n}\) | Real Multiplication Distributes over Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \sequence {\lambda \cdot a_n} +_\R \sequence {\mu \cdot a_n}\) | 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 |
$\Box$
Distributivity over Vector Addition
\(\ds \lambda \times_\R \sequence {a_n + b_n}\) | \(=\) | \(\ds \lambda \times_\R \paren {\sequence {a_n} +_\R \sequence {b_n} }\) | Definition of Pointwise Addition on Ring of Sequences | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \times_\R \sequence {a_n} +_\R \lambda \times_\R \sequence {b_n}\) | Real Multiplication Distributes over Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \sequence {\lambda \cdot a_n} +_\R \sequence {\lambda \cdot b_n}\) | 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
\(\ds \sequence {\paren {\lambda \times_\R \mu} \cdot a_n}\) | \(=\) | \(\ds \paren {\lambda \times_\R \mu} \times_\R \sequence {a_n}\) | Definition of Pointwise Scalar Multiplication on Ring of Sequences | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \times_\R \paren {\mu \times_\R \sequence {a_n} }\) | Real Multiplication is Associative | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \times_\R \sequence {\mu \cdot a_n}\) | 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
\(\ds \sequence {1 \cdot a_n}\) | \(=\) | \(\ds 1 \times_\R \sequence {a_n}\) | Definition of Pointwise Scalar Multiplication on Ring of Sequences | |||||||||||
\(\ds \) | \(=\) | \(\ds \sequence {a_n}\) |
$\Box$
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.1$: Vector Space