Convergence Theory
Disclaimer: These are my personal notes compiled for my own reference and learning. They may contain errors, incomplete information, or personal interpretations. While I strive for accuracy, these notes are not peer-reviewed and should not be considered authoritative sources. Please consult official textbooks, research papers, or other reliable sources for academic or professional purposes.
1. Pointwise Convergence
1.1 Definition
A sequence of functions $\{f_n\}$ converges pointwise to $f$ on a set $E$ if:
$$\forall x \in E, \forall \epsilon > 0, \exists N(x, \epsilon) \text{ such that } n \geq N \Rightarrow |f_n(x) - f(x)| < \epsilon$$
Note that $N$ may depend on both $x$ and $\epsilon$.
1.2 Properties
- Limit function: $f(x) = \lim_{n \to \infty} f_n(x)$ for each $x \in E$
- Preservation of continuity: Not guaranteed (pointwise limit of continuous functions may be discontinuous)
- Integration: $\lim_{n \to \infty} \int f_n$ may not equal $\int \lim_{n \to \infty} f_n$
1.3 Example
Consider $f_n(x) = x^n$ on $[0,1]$:
$$f(x) = \lim_{n \to \infty} x^n = \begin{cases}
0 & \text{if } 0 \leq x < 1 \\
1 & \text{if } x = 1
\end{cases}$$
Each $f_n$ is continuous, but $f$ is discontinuous at $x = 1$.
2. Uniform Convergence
2.1 Definition
A sequence of functions $\{f_n\}$ converges uniformly to $f$ on $E$ if:
$$\forall \epsilon > 0, \exists N(\epsilon) \text{ such that } n \geq N \Rightarrow \sup_{x \in E} |f_n(x) - f(x)| < \epsilon$$
Note that $N$ depends only on $\epsilon$, not on $x$.
2.2 Equivalent Characterization
$f_n \to f$ uniformly on $E$ if and only if:
$$\lim_{n \to \infty} \sup_{x \in E} |f_n(x) - f(x)| = 0$$
2.3 Weierstrass M-Test
If $\sum M_n$ converges and $|f_n(x)| \leq M_n$ for all $x \in E$, then $\sum f_n$ converges uniformly on $E$.
3. Properties of Uniform Convergence
3.1 Continuity Preservation
Theorem: If $f_n \to f$ uniformly on $E$ and each $f_n$ is continuous at $x_0 \in E$, then $f$ is continuous at $x_0$.
3.2 Integration
Theorem: If $f_n \to f$ uniformly on $[a,b]$ and each $f_n$ is integrable, then:
$$\lim_{n \to \infty} \int_a^b f_n(x) dx = \int_a^b f(x) dx$$
3.3 Differentiation
Theorem: If $f_n \to f$ pointwise on $[a,b]$, each $f_n$ is differentiable, and $f_n' \to g$ uniformly on $[a,b]$, then $f$ is differentiable and $f' = g$.
4. Types of Convergence for Function Series
4.1 Pointwise Convergence of Series
$\sum_{n=1}^{\infty} f_n(x)$ converges pointwise to $S(x)$ if the sequence of partial sums $S_N(x) = \sum_{n=1}^N f_n(x)$ converges pointwise to $S(x)$.
4.2 Uniform Convergence of Series
$\sum_{n=1}^{\infty} f_n(x)$ converges uniformly if the partial sums converge uniformly.
4.3 Absolute Convergence
$\sum f_n(x)$ converges absolutely at $x$ if $\sum |f_n(x)|$ converges.
5. Modes of Convergence for Sequences
5.1 Almost Everywhere Convergence
$f_n \to f$ almost everywhere (a.e.) if there exists a set $E$ of measure zero such that $f_n(x) \to f(x)$ for all $x \notin E$.
5.2 Convergence in Measure
$f_n \to f$ in measure if:
$$\forall \epsilon > 0: \lim_{n \to \infty} \mu(\{x : |f_n(x) - f(x)| \geq \epsilon\}) = 0$$
5.3 $L^p$ Convergence
$f_n \to f$ in $L^p$ if:
$$\lim_{n \to \infty} \int |f_n - f|^p d\mu = 0$$
6. Relationships Between Convergence Types
6.1 Hierarchy of Convergence
- Uniform convergence $\Rightarrow$ Pointwise convergence
- Uniform convergence $\Rightarrow$ Convergence in measure
- $L^p$ convergence $\Rightarrow$ Convergence in measure
- Convergence in measure $\Rightarrow$ Almost everywhere convergence (subsequence)
6.2 Counterexamples
- Pointwise but not uniform: $f_n(x) = x^n$ on $[0,1]$
- $L^1$ but not pointwise: Moving bump functions
- Almost everywhere but not in measure: Functions with increasing support
7. Uniform Convergence Tests
7.1 Cauchy Criterion
$\{f_n\}$ converges uniformly on $E$ if and only if:
$$\forall \epsilon > 0, \exists N \text{ such that } m,n \geq N \Rightarrow \sup_{x \in E} |f_m(x) - f_n(x)| < \epsilon$$
7.2 Dini's Theorem
If $f_n \to f$ pointwise on a compact set $K$, each $f_n$ is continuous, $f$ is continuous, and the convergence is monotonic, then the convergence is uniform.
7.3 Arzela-Ascoli Theorem
A family of functions on a compact set is compact if and only if it is equicontinuous and uniformly bounded.
8. Stone-Weierstrass Theorem
8.1 Statement
Let $X$ be a compact metric space and $\mathcal{A}$ be an algebra of continuous real-valued functions on $X$ that:
- Separates points of $X$
- Contains the constant functions
Then $\mathcal{A}$ is dense in $C(X)$ with respect to the uniform norm.
8.2 Applications
- Polynomial approximation: Polynomials are dense in $C[a,b]$
- Trigonometric approximation: Trigonometric polynomials are dense in continuous $2\pi$-periodic functions
9. Convergence in Normed Spaces
9.1 Strong Convergence
In a normed space $(X, \|\cdot\|)$, $x_n \to x$ strongly if $\|x_n - x\| \to 0$.
9.2 Weak Convergence
$x_n \to x$ weakly if $f(x_n) \to f(x)$ for all $f \in X^*$ (the dual space).
9.3 Weak* Convergence
In the dual space $X^*$, $f_n \to f$ weak* if $f_n(x) \to f(x)$ for all $x \in X$.
10. Banach-Steinhaus Theorem
Uniform Boundedness Principle: Let $\{T_n\}$ be a sequence of bounded linear operators from a Banach space $X$ to a normed space $Y$. If $\sup_n \|T_n x\| < \infty$ for each $x \in X$, then $\sup_n \|T_n\| < \infty$.
11. Dominated Convergence Theorem
If $f_n \to f$ almost everywhere, $|f_n| \leq g$ where $g$ is integrable, then:
$$\lim_{n \to \infty} \int f_n d\mu = \int f d\mu$$
12. Code Examples
# Python implementations for convergence analysis
import numpy as np
import matplotlib.pyplot as plt
from scipy import integrate
import warnings
warnings.filterwarnings('ignore')
class ConvergenceAnalyzer:
def __init__(self, function_sequence, domain=None, true_limit=None):
"""
Analyze convergence of function sequences
function_sequence: list of functions or callable that takes n and returns function
domain: array of x values to test
true_limit: known limit function for comparison
"""
self.function_sequence = function_sequence
self.domain = domain if domain is not None else np.linspace(0, 1, 100)
self.true_limit = true_limit
def get_function(self, n):
"""Get the nth function in the sequence"""
if callable(self.function_sequence):
return self.function_sequence(n)
else:
return self.function_sequence[n]
def pointwise_convergence_test(self, n_terms=50, test_points=None):
"""Test pointwise convergence at specific points"""
if test_points is None:
test_points = [0.25, 0.5, 0.75]
results = {}
for x in test_points:
values = []
for n in range(1, n_terms + 1):
fn = self.get_function(n)
try:
value = fn(x)
values.append(value)
except:
values.append(np.nan)
results[x] = values
# Check for convergence
if len(values) > 10:
tail = values[-10:]
if not any(np.isnan(tail)):
variance = np.var(tail)
if variance < 1e-10:
limit_value = np.mean(tail)
print(f"Point x = {x}: Appears to converge to {limit_value:.6f}")
else:
print(f"Point x = {x}: No clear convergence (variance = {variance:.2e})")
return results
def uniform_convergence_test(self, n_terms=50):
"""Test uniform convergence using supremum norm"""
if self.true_limit is None:
print("True limit function needed for uniform convergence test")
return None
supremum_errors = []
for n in range(1, n_terms + 1):
fn = self.get_function(n)
# Compute |f_n(x) - f(x)| for all x in domain
errors = []
for x in self.domain:
try:
error = abs(fn(x) - self.true_limit(x))
errors.append(error)
except:
errors.append(np.inf)
sup_error = max(errors) if errors else np.inf
supremum_errors.append(sup_error)
# Check if supremum errors approach 0
if len(supremum_errors) > 10:
tail = supremum_errors[-10:]
if all(e < 1e-6 for e in tail):
print("Uniform convergence detected")
else:
print(f"No uniform convergence (final sup error: {tail[-1]:.2e})")
return supremum_errors
def plot_convergence(self, n_values=[1, 5, 10, 20, 50], show_limit=True):
"""Plot function sequence and limit"""
plt.figure(figsize=(12, 8))
colors = plt.cm.viridis(np.linspace(0, 0.8, len(n_values)))
for i, n in enumerate(n_values):
fn = self.get_function(n)
y_values = [fn(x) for x in self.domain]
plt.plot(self.domain, y_values, color=colors[i],
linewidth=2, label=f'f_{n}(x)', alpha=0.7)
if show_limit and self.true_limit is not None:
limit_values = [self.true_limit(x) for x in self.domain]
plt.plot(self.domain, limit_values, 'r-',
linewidth=3, label='Limit function', alpha=0.9)
plt.xlabel('x')
plt.ylabel('f_n(x)')
plt.title('Function Sequence Convergence')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()
def plot_supremum_errors(self, n_terms=50):
"""Plot supremum errors to visualize uniform convergence"""
sup_errors = self.uniform_convergence_test(n_terms)
if sup_errors is not None:
plt.figure(figsize=(10, 6))
plt.semilogy(range(1, len(sup_errors) + 1), sup_errors, 'bo-', linewidth=2)
plt.xlabel('n')
plt.ylabel('sup |f_n(x) - f(x)|')
plt.title('Supremum Error (Log Scale)')
plt.grid(True, alpha=0.3)
plt.show()
class SeriesConvergenceAnalyzer:
def __init__(self, term_functions, domain=None):
"""
Analyze convergence of function series
term_functions: list of functions or callable returning nth term function
"""
self.term_functions = term_functions
self.domain = domain if domain is not None else np.linspace(-1, 1, 100)
def get_term(self, n):
"""Get nth term function"""
if callable(self.term_functions):
return self.term_functions(n)
else:
return self.term_functions[n]
def weierstrass_m_test(self, n_terms=50):
"""Apply Weierstrass M-test for uniform convergence"""
M_values = []
for n in range(n_terms):
fn = self.get_term(n)
# Find supremum of |f_n(x)| over domain
max_val = 0
for x in self.domain:
try:
val = abs(fn(x))
max_val = max(max_val, val)
except:
max_val = np.inf
break
M_values.append(max_val)
# Check if sum of M_n converges
partial_sums = np.cumsum(M_values)
print("Weierstrass M-Test:")
print(f"M_values (first 10): {M_values[:10]}")
print(f"Partial sum of M_n: {partial_sums[-1]:.6f}")
if partial_sums[-1] < np.inf and len(M_values) > 10:
tail_growth = partial_sums[-1] - partial_sums[-11]
if tail_growth < 1e-6:
print("M-test suggests uniform convergence")
else:
print("M-test inconclusive")
return M_values, partial_sums
def plot_partial_sums(self, n_terms=[5, 10, 20, 50]):
"""Plot partial sums of the series"""
plt.figure(figsize=(12, 8))
colors = plt.cm.plasma(np.linspace(0, 0.8, len(n_terms)))
for i, N in enumerate(n_terms):
# Compute partial sum S_N(x)
partial_sum = np.zeros_like(self.domain)
for n in range(N):
fn = self.get_term(n)
for j, x in enumerate(self.domain):
try:
partial_sum[j] += fn(x)
except:
partial_sum[j] = np.nan
plt.plot(self.domain, partial_sum, color=colors[i],
linewidth=2, label=f'S_{N}(x)', alpha=0.8)
plt.xlabel('x')
plt.ylabel('S_N(x)')
plt.title('Partial Sums of Function Series')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()
def demonstrate_convergence_types():
"""Demonstrate different types of convergence with examples"""
print("=== Convergence Types Demonstration ===\n")
# Example 1: f_n(x) = x^n on [0,1] - pointwise but not uniform
print("1. f_n(x) = x^n on [0,1]")
print(" Pointwise convergent but not uniformly convergent")
def power_sequence(n):
return lambda x: x**n
def power_limit(x):
return 0 if x < 1 else 1
analyzer1 = ConvergenceAnalyzer(power_sequence,
domain=np.linspace(0, 1, 100),
true_limit=power_limit)
print("Pointwise convergence test:")
analyzer1.pointwise_convergence_test(n_terms=20, test_points=[0.5, 0.9, 1.0])
print("\nUniform convergence test:")
analyzer1.uniform_convergence_test(n_terms=20)
# Example 2: f_n(x) = x/n - uniformly convergent
print("\n2. f_n(x) = x/n")
print(" Uniformly convergent to f(x) = 0")
def linear_sequence(n):
return lambda x: x/n
analyzer2 = ConvergenceAnalyzer(linear_sequence,
domain=np.linspace(-1, 1, 100),
true_limit=lambda x: 0)
analyzer2.uniform_convergence_test(n_terms=50)
# Example 3: Fourier series convergence
print("\n3. Fourier Series: f(x) = x on [-π, π]")
print(" Pointwise convergent (except at discontinuities)")
def fourier_partial_sum(N):
def partial_sum(x):
result = 0
for n in range(1, N+1):
result += (2 * (-1)**(n+1) / n) * np.sin(n * x)
return result
return partial_sum
domain_fourier = np.linspace(-np.pi, np.pi, 200)
analyzer3 = ConvergenceAnalyzer(fourier_partial_sum,
domain=domain_fourier,
true_limit=lambda x: x)
analyzer3.pointwise_convergence_test(n_terms=20, test_points=[0, np.pi/2, np.pi])
def power_series_convergence():
"""Analyze convergence of power series"""
print("\n=== Power Series Convergence ===")
# Geometric series: sum(x^n) for |x| < 1
def geometric_term(n):
return lambda x: x**n
domain = np.linspace(-0.9, 0.9, 100) # Inside radius of convergence
series_analyzer = SeriesConvergenceAnalyzer(geometric_term, domain)
print("Geometric series: Σ x^n")
M_vals, partial_sums = series_analyzer.weierstrass_m_test(20)
# Plot convergence
series_analyzer.plot_partial_sums([5, 10, 15, 20])
def demonstrate_dinis_theorem():
"""Demonstrate Dini's theorem"""
print("\n=== Dini's Theorem Demonstration ===")
# Monotone sequence of continuous functions on compact set
def dini_sequence(n):
return lambda x: x**(1 + 1/n)
domain = np.linspace(0, 1, 100)
analyzer = ConvergenceAnalyzer(dini_sequence, domain, lambda x: x)
print("f_n(x) = x^(1 + 1/n) on [0,1]")
print("Monotone decreasing sequence of continuous functions")
print("Dini's theorem guarantees uniform convergence")
sup_errors = analyzer.uniform_convergence_test(50)
if sup_errors:
print(f"Final supremum error: {sup_errors[-1]:.2e}")
# Example usage
if __name__ == "__main__":
demonstrate_convergence_types()
power_series_convergence()
demonstrate_dinis_theorem()
# Additional visualization examples
print("\n=== Visualization Examples ===")
# Create some interesting convergence examples
# Example: sin(nx)/n
def sinusoidal_sequence(n):
return lambda x: np.sin(n*x)/n
domain = np.linspace(0, 2*np.pi, 300)
analyzer = ConvergenceAnalyzer(sinusoidal_sequence, domain, lambda x: 0)
print("\nf_n(x) = sin(nx)/n - uniformly convergent to 0")
analyzer.plot_convergence([1, 2, 5, 10, 20])
analyzer.plot_supremum_errors(30)
13. Applications
13.1 Functional Analysis
- Banach and Hilbert space theory
- Operator theory and spectral analysis
- Fixed point theorems
13.2 Partial Differential Equations
- Method of separation of variables
- Fourier analysis and transforms
- Weak solutions and distribution theory
13.3 Numerical Analysis
- Approximation theory
- Finite element methods
- Iterative algorithms convergence
13.4 Probability Theory
- Law of large numbers
- Central limit theorem
- Martingale convergence theorems
14. References
- Rudin, W. (1976). Principles of Mathematical Analysis.
- Royden, H. L., & Fitzpatrick, P. M. (2010). Real Analysis.
- Folland, G. B. (1999). Real Analysis: Modern Techniques and Their Applications.