Inner product spaces give us powerful tools to measure vector lengths and angles. Norms and distances, derived from inner products, let us quantify these concepts mathematically. This opens up a whole new world of geometric intuition in abstract vector spaces.

These ideas are crucial for understanding the structure of inner product spaces. We'll explore how norms relate to metrics, dive into orthogonality and projections, and see how these concepts shape the geometry of these spaces.

Norms induced by inner products

Definition and properties of induced norms

• Induced norm defined as ||x|| = √⟨x,x⟩ for any vector x in vector space V
• Satisfies non-negativity, positive definiteness, homogeneity, and triangle inequality properties
• Represents length or magnitude of a vector in real inner product spaces
• Euclidean norm on ℝⁿ derived from standard dot product (special case)
• Computation involves evaluating inner product of vector with itself and taking square root
• Cauchy-Schwarz inequality relates inner product of two vectors to product of induced norms: |⟨x,y⟩| ≤ ||x|| ||y||
• Parallelogram law states ||x+y||² + ||x-y||² = 2(||x||² + ||y||²) for any vectors x and y

Examples and applications

• Calculate induced norm for vector (3, 4) in ℝ² using standard dot product
• Compute induced norm for complex vector (1+i, 2-i) in ℂ² with standard inner product
• Apply Cauchy-Schwarz inequality to estimate inner product of vectors (1, 2, 3) and (4, 5, 6)
• Verify parallelogram law for vectors (1, 1) and (2, -1) in ℝ²
• Use induced norm to find length of polynomial 2x² + 3x + 1 in space of polynomials with degree ≤ 2

Triangle inequality for norms

Proof strategy and key steps

• Triangle inequality states ||x+y|| ≤ ||x|| + ||y|| for any vectors x and y
• Proof relies on properties of inner products and Cauchy-Schwarz inequality
• Expand ||x+y||² using definition of induced norm
• Apply Cauchy-Schwarz inequality to cross-terms
• Demonstrate square of left-hand side ≤ square of right-hand side
• Take square root of both sides, preserving inequality due to monotonicity of square root function

Implications and applications

• Establishes crucial property for norms, essential for defining metric spaces
• Allows estimation of norm of sum of vectors based on individual norms
• Used in error analysis and approximation theory (bounding errors in vector addition)
• Applies in signal processing for analyzing combined signals
• Generalizes to infinite-dimensional spaces (functional analysis)

Norms vs Metrics

Relationship between norms and metrics

• Norm induces metric through formula d(x,y) = ||x-y||
• Induced metric satisfies non-negativity, symmetry, positive definiteness, and triangle inequality
• Bijective relationship exists between norms and translation-invariant, homogeneous metrics
• Completeness in normed spaces defined using induced metric (Banach spaces)
• Topology of normed space determined by induced metric
• Equivalence of norms on finite-dimensional spaces implies same topology

Examples and applications

• Derive Manhattan metric from L1 norm in ℝⁿ
• Show Euclidean metric arises from L2 norm
• Demonstrate how max norm induces Chebyshev metric
• Use induced metric to define open and closed sets in normed vector spaces
• Apply concept of completeness to show ℝⁿ with Euclidean norm is complete

Geometry of inner product spaces

Orthogonality and projections

• Orthogonality defined using inner product ⟨x,y⟩ = 0
• Pythagorean theorem states ||x+y||² = ||x||² + ||y||² for orthogonal vectors x and y
• Angle between vectors defined as cos θ = ⟨x,y⟩ / (||x|| ||y||)
• Orthogonal decomposition theorem allows vector representation as sum of projection onto subspace and orthogonal vector
• Gram-Schmidt process constructs orthonormal basis from linearly independent set

Isometries and convexity

• Isometries preserve distances and norms (linear transformations)
• Play crucial role in understanding geometry of inner product spaces
• Examples include rotations, reflections, and orthogonal transformations
• Convexity in inner product spaces relies on properties of norms and distances
• Leads to important results in optimization theory (convex optimization)
• Applications include finding minimum distance between point and subspace