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
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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