Functional Analysis

Functional analysis studies vector spaces endowed with a limit-related structure (like a metric or topology) and the linear operators acting upon them.

1. Normed Linear Spaces

A Normed Linear Space $(V, \|\cdot\|)$ is a vector space $V$ over a field (usually $\mathbb{C}$ or $\mathbb{R}$) with a norm function $\|\cdot\|: V \to [0, \infty)$ satisfying:

1. Definiteness: $\|v\| = 0 \iff v = 0$.
2. Homogeneity: $\|\alpha v\| = |\alpha| \cdot \|v\|$.
3. Triangle Inequality: $\|u + v\| \le \|u\| + \|v\|$.

A Banach Space is a normed linear space that is complete (every Cauchy sequence converges).

2. Hilbert Spaces and Inner Products

A Hilbert Space is a complete inner product space. The inner product $\langle \cdot, \cdot \rangle$ satisfies:

1. Linearity: $\langle \alpha u + \beta v, w \rangle = \alpha \langle u, w \rangle + \beta \langle v, w \rangle$.
2. Conjugate Symmetry: $\langle u, v \rangle = \overline{\langle v, u \rangle}$.
3. Positive Definiteness: $\langle v, v \rangle \ge 0$, with equality only if $v=0$.

The $L^2$ Space is a primary example, where functions are square-integrable: $\langle f, g \rangle = \int_a^b f(x) \overline{g(x)} \, dx, \quad \|f\|_2 = \sqrt{\int_a^b |f(x)|^2 \, dx}$

import numpy as np

# Numerical verification of the Triangle Inequality
# ||f + g|| <= ||f|| + ||g||
x = np.linspace(0, 1, 500)
f = x**2
g = np.exp(x)

def norm_l2(y, dx):
  return np.sqrt(np.trapz(y**2, dx))

dx = x[1] - x[0]
n_f = norm_l2(f, x)
n_g = norm_l2(g, x)
n_sum = norm_l2(f + g, x)

print(f"||f||: {n_f:.4f}")
print(f"||g||: {n_g:.4f}")
print(f"||f+g||: {n_sum:.4f}")
print(f"Is {n_sum:.4f} <= {n_f + n_g:.4f}? {n_sum <= n_f + n_g + 1e-9}")

3. Linear Operators

1. Partial Differential Equations: Proving that a solution exists even if we can’t find it.
2. Signal Processing: Decomposing complex signals into simple frequencies.
3. Quantum Mechanics: The state of a particle is a vector in an infinite-dimensional Hilbert space.

Exercises