LA 101
M06 · L02
Module 6: Inner Product Spaces

Function Spaces

Functions can be added and scaled just like vectors. Equip the resulting space with an inner product and you get infinite-dimensional geometry — the language of Fourier analysis and signal processing.

01 / 10
LA 101
M06 · L02
Functions Are Vectors

The Vector Space of Functions

C([a, b]) satisfies all ten vector space axioms. Addition is pointwise: (f+g)(t) = f(t)+g(t). Scaling is pointwise: (αf)(t) = αf(t). The zero vector is the zero function.

∞
dimensions
10
axioms hold
L²
key space
02 / 10
LA 101
M06 · L02
The Energy Space

The L² Space

L²([a, b]) contains all functions with finite energy:

L² Definition
\int_a^b |f(t)|^2\,dt < \infty
Why L²?
Physical signals have finite power. The condition ∫f²dt < ∞ directly captures this physical constraint.
03 / 10
LA 101
M06 · L02
Complete Inner Product Space

The Hilbert Space

A Hilbert space is an inner product space that is also complete — every Cauchy sequence converges inside the space.

  • ℝⁿ with dot product — simplest Hilbert space
  • L²([a, b]) — infinite-dimensional Hilbert space
  • C([a, b]) — inner product space, but NOT complete
  • Completeness = closed arena for infinite-dim analysis
04 / 10
LA 101
M06 · L02
Infinite-Dimensional Coordinates

Generalized Fourier Series

Expansion
f = \sum_{n=1}^{\infty}\langle f,\varphi_n\rangle\,\varphi_n
Same as ℝⁿ
The coefficients cₙ = ⟨f, φₙ⟩ are inner products — exactly like extracting coordinates via dot products.
05 / 10
LA 101
M06 · L02
Energy Conservation

Parseval's Theorem

Parseval
\|f\|^2 = \int_a^b|f|^2\,dt = \sum_{n=1}^{\infty}|c_n|^2

The infinite-dimensional Pythagorean theorem. Time-domain energy equals the sum of squared Fourier coefficients — energy is preserved in the transform.

06 / 10
LA 101
M06 · L02
Orthogonal Families

Beyond Sines & Cosines

  • Trigonometric system — L²([0, 2π]), classical Fourier
  • Legendre polynomials — L²([−1, 1]), spherical symmetry
  • Hermite polynomials — weighted L²(ℝ), Gaussian channels
  • Wavelets — time–frequency localization, compression
  • Each family is the natural basis for its specific space
07 / 10
LA 101
Knowledge Check

Check what stuck

Four questions from this lesson. Answer to see why — the explanation appears whether you were right or wrong. Nothing is scored or saved.

Question 1 of 0
Score 0/0

08 / 10
LA 101
M06 · L02
Engineering Applications

Function Spaces in Practice

  • OFDM — project received signal onto N orthogonal subcarriers (DFT)
  • Sampling theorem — bandlimited subspace, Parseval enables reconstruction
  • Lowpass filtering — orthogonal projection onto bandlimited subspace
  • Channel estimation — expand channel in Doppler-delay basis
  • Wavelet denoising — sparse representation in wavelet basis
09 / 10
LA 101
Up Next
Lesson 2 Complete

The Infinite Frontier!

Functions live in Hilbert spaces. Orthonormal bases unlock coordinate-free expansion. Parseval conserves energy. Projection gives best approximation. This is linear algebra — just in infinite dimensions.

Coming Up
M6-L3: Fourier Series Connection →
All lessons Read article
10 / 10