LA 101
M06 · L01
Module 6: Inner Product Spaces

Inner Products

The dot product is a special case of a broader structure. Any operation satisfying four axioms lets you measure length, angle, and orthogonality — even between functions.

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LA 101
M06 · L01
The Abstraction

What is an Inner Product?

A function ⟨·,·⟩ : V × V → ℝ that generalizes the dot product. Fixing the formula is not what matters — the four axioms are.

⟨u,v⟩
notation
4
axioms
∞
spaces
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LA 101
M06 · L01
The Four Axioms

Rules of the Game

  • Linearity — ⟨αu+v, w⟩ = α⟨u,w⟩ + ⟨v,w⟩
  • Symmetry — ⟨u,v⟩ = ⟨v,u⟩
  • Non-negativity — ⟨v,v⟩ ≥ 0
  • Definiteness — ⟨v,v⟩ = 0 iff v = 0
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LA 101
M06 · L01
Length from Structure

The Induced Norm

Every inner product automatically gives a way to measure length — the induced norm:

Induced Norm
\|\mathbf{v}\| = \sqrt{\langle \mathbf{v},\, \mathbf{v} \rangle}
Note
Not every norm comes from an inner product. A norm does iff it satisfies the parallelogram law.
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LA 101
M06 · L01
The Master Inequality

Cauchy-Schwarz

Cauchy-Schwarz
|\langle \mathbf{u},\, \mathbf{v} \rangle| \leq \|\mathbf{u}\|\,\|\mathbf{v}\|
Why it matters
Guarantees that |⟨u,v⟩| / (‖u‖‖v‖) ≤ 1, so the cosine formula for angle is valid in any inner product space.
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LA 101
M06 · L01
Geometry in Any Space

Angle and Orthogonality

Angle Definition
\cos\theta = \dfrac{\langle \mathbf{u},\, \mathbf{v} \rangle}{\|\mathbf{u}\|\,\|\mathbf{v}\|}

Two vectors are orthogonal when ⟨u,v⟩ = 0 (θ = 90°). This definition works for polynomials, matrices, signals — any inner product space.

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LA 101
M06 · L01
Functions as Vectors

The L² Inner Product

Functions on [a, b] form a vector space. The standard inner product is:

L² Inner Product
\langle f,\, g \rangle = \int_a^b f(t)\,g(t)\,dt

Sine and cosine of the same frequency are orthogonal under this inner product on [0, 2π] — the foundation of Fourier series.

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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
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LA 101
M06 · L01
Signal Processing

Inner Products in Practice

  • Matched filtering — ∫r(t)s(t)dt maximizes SNR (Cauchy-Schwarz)
  • Correlation — inner product as a function of time lag
  • Fourier coefficients — projections onto complex exponentials
  • OFDM subcarriers — orthogonal by L² inner product
  • Frobenius norm — inner product on matrices, used in PCA
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LA 101
Up Next
Lesson 1 Complete

Module 6 Begins!

You now have the abstract framework: four axioms give you length, angle, and orthogonality in any vector space — including the function spaces at the heart of signal processing.

Coming Up
M6-L2: Function Spaces →
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