Questions: Inner Product Spaces

Question 1

Which inner product axiom guarantees that ⟨v, v⟩ = 0 implies v = 0?

• A. Symmetry
• B. Bilinearity
• C. Positive definiteness
• D. The Cauchy-Schwarz inequality

Answer: (C)

Positive definiteness states that ⟨v, v⟩ ≥ 0 for all v, with equality if and only if v = 0. This is what makes the induced norm ‖v‖ = √⟨v,v⟩ a true norm — it ensures only the zero vector has zero length. Without positive definiteness, the 'norm' could vanish for nonzero vectors, breaking the geometric interpretation.

Question 2

Most inner product on a vector space is just the standard dot product.

• T. True
• F. False

Answer: False

The dot product is the standard inner product on ℝⁿ, but many other valid inner products exist on the same or different spaces. For example, on the space of continuous functions on [a, b], the integral ⟨f, g⟩ = ∫_a^b f(x)g(x) dx defines an inner product. Different inner products on the same space define different notions of length and orthogonality, and the 'right' inner product depends on the application.

Question 3

How does an inner product define orthogonality in an abstract vector space where there is no visual geometry?

Model answer: Two vectors u and v are defined to be orthogonal if ⟨u, v⟩ = 0. The inner product axioms are chosen precisely so that this algebraic condition generalizes the geometric notion of perpendicularity from ℝ² and ℝ³ to any vector space.

In ℝ², perpendicular vectors have dot product zero — this is a theorem provable from geometry. In an abstract vector space with an inner product, orthogonality is defined by the same algebraic condition. This allows concepts like orthogonal projections, Gram-Schmidt orthogonalization, and Fourier series to work in function spaces and other abstract settings.