Convergence in normed linear spaces is a key concept in functional analysis. It defines how sequences of elements approach a limit point , using the norm to measure distance between elements.
Uniqueness of limits and Cauchy sequences are crucial ideas in this context. These concepts help us understand completeness , which ensures that every Cauchy sequence converges within the space, leaving no "gaps."
Convergence in normed linear spaces
Convergence in normed linear spaces
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Defines convergence of a sequence ( x n ) (x_n) ( x n ) in a normed linear space ( X , ∥ ⋅ ∥ ) (X, \|\cdot\|) ( X , ∥ ⋅ ∥ ) as lim n → ∞ ∥ x n − x ∥ = 0 \lim_{n \to \infty} \|x_n - x\| = 0 lim n → ∞ ∥ x n − x ∥ = 0 for some x ∈ X x \in X x ∈ X
Denotes convergence as x n → x x_n \to x x n → x or lim n → ∞ x n = x \lim_{n \to \infty} x_n = x lim n → ∞ x n = x
Illustrates convergence using the space of continuous functions C [ 0 , 1 ] C[0,1] C [ 0 , 1 ] with the supremum norm ∥ f ∥ ∞ = sup x ∈ [ 0 , 1 ] ∣ f ( x ) ∣ \|f\|_\infty = \sup_{x \in [0,1]} |f(x)| ∥ f ∥ ∞ = sup x ∈ [ 0 , 1 ] ∣ f ( x ) ∣
Defines a sequence of functions ( f n ) (f_n) ( f n ) by f n ( x ) = x n f_n(x) = x^n f n ( x ) = x n for x ∈ [ 0 , 1 ] x \in [0,1] x ∈ [ 0 , 1 ]
Shows ( f n ) (f_n) ( f n ) converges to the function f ( x ) = 0 f(x) = 0 f ( x ) = 0 for x ∈ [ 0 , 1 ) x \in [0,1) x ∈ [ 0 , 1 ) and f ( 1 ) = 1 f(1) = 1 f ( 1 ) = 1
Uniqueness of limits
Assumes ( x n ) (x_n) ( x n ) converges to both x x x and y y y in a normed space ( X , ∥ ⋅ ∥ ) (X, \|\cdot\|) ( X , ∥ ⋅ ∥ )
Applies the triangle inequality to obtain ∥ x − y ∥ ≤ ∥ x − x n ∥ + ∥ x n − y ∥ \|x - y\| \leq \|x - x_n\| + \|x_n - y\| ∥ x − y ∥ ≤ ∥ x − x n ∥ + ∥ x n − y ∥
Takes the limit as n → ∞ n \to \infty n → ∞ , yielding lim n → ∞ ∥ x − x n ∥ = 0 \lim_{n \to \infty} \|x - x_n\| = 0 lim n → ∞ ∥ x − x n ∥ = 0 and lim n → ∞ ∥ x n − y ∥ = 0 \lim_{n \to \infty} \|x_n - y\| = 0 lim n → ∞ ∥ x n − y ∥ = 0
Concludes ∥ x − y ∥ ≤ 0 + 0 = 0 \|x - y\| \leq 0 + 0 = 0 ∥ x − y ∥ ≤ 0 + 0 = 0 , implying x = y x = y x = y
Proves the uniqueness of limits in normed spaces
Cauchy sequences and completeness
Defines a Cauchy sequence ( x n ) (x_n) ( x n ) in a normed space ( X , ∥ ⋅ ∥ ) (X, \|\cdot\|) ( X , ∥ ⋅ ∥ ) as one where for every ε > 0 \varepsilon > 0 ε > 0 , there exists N ∈ N N \in \mathbb{N} N ∈ N such that ∥ x m − x n ∥ < ε \|x_m - x_n\| < \varepsilon ∥ x m − x n ∥ < ε for all m , n ≥ N m, n \geq N m , n ≥ N
Explains intuitively that the terms of a Cauchy sequence become arbitrarily close to each other as the sequence progresses
Relates Cauchy sequences to completeness by stating a normed space ( X , ∥ ⋅ ∥ ) (X, \|\cdot\|) ( X , ∥ ⋅ ∥ ) is complete if every Cauchy sequence in X X X converges to an element of X X X
Emphasizes completeness ensures there are no "gaps" in the space, and every Cauchy sequence has a limit within the space
Completeness via Cauchy criterion
Outlines proving a normed space ( X , ∥ ⋅ ∥ ) (X, \|\cdot\|) ( X , ∥ ⋅ ∥ ) is complete by showing every Cauchy sequence in X X X converges to an element of X X X
Demonstrates completeness of ( R , ∣ ⋅ ∣ ) (\mathbb{R}, |\cdot|) ( R , ∣ ⋅ ∣ ) as an example
Lets ( x n ) (x_n) ( x n ) be a Cauchy sequence in R \mathbb{R} R
Applies the Cauchy criterion to obtain for every ε > 0 \varepsilon > 0 ε > 0 , there exists N ∈ N N \in \mathbb{N} N ∈ N such that ∣ x m − x n ∣ < ε |x_m - x_n| < \varepsilon ∣ x m − x n ∣ < ε for all m , n ≥ N m, n \geq N m , n ≥ N
Uses the completeness of R \mathbb{R} R as an ordered field to show ( x n ) (x_n) ( x n ) converges to a real number x x x
Concludes ( R , ∣ ⋅ ∣ ) (\mathbb{R}, |\cdot|) ( R , ∣ ⋅ ∣ ) is complete