4.2 Norm and distance in inner product spaces
3 min read•august 16, 2024
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 defined as ||x|| = √⟨x,x⟩ for any vector x in vector space V
- Satisfies non-negativity, , homogeneity, and properties
- Represents length or magnitude of a vector in real inner product spaces
- on ℝⁿ derived from standard dot product (special case)
- Computation involves evaluating inner product of vector with itself and taking square root
- relates inner product of two vectors to product of induced norms: |⟨x,y⟩| ≤ ||x|| ||y||
- 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
Key Terms to Review (16)
Cauchy sequence: A Cauchy sequence is a sequence of elements in a metric space where, for every positive distance, there exists a point in the sequence beyond which all terms are within that distance of each other. This concept emphasizes the idea that as you progress further along in the sequence, the terms become arbitrarily close to one another, indicating that the sequence converges to a limit. In inner product spaces, this relates closely to the notions of norms and distances, where the properties of Cauchy sequences help establish whether or not a space is complete.
Cauchy-Schwarz Inequality: The Cauchy-Schwarz inequality states that for any vectors $\mathbf{u}$ and $\mathbf{v}$ in an inner product space, the absolute value of their inner product is less than or equal to the product of their norms. This can be expressed mathematically as $|\langle \mathbf{u}, \mathbf{v} \rangle| \leq ||\mathbf{u}|| ||\mathbf{v}||$. This inequality is foundational in understanding concepts such as linear independence, orthogonality, and measuring distances in vector spaces, making it crucial for analyzing relationships between vectors and their properties in higher dimensions.
Convergent Sequence: A convergent sequence is a sequence of elements in a space that approaches a specific limit as the index progresses to infinity. In inner product spaces, this concept is tied closely to the notion of distance and norms, as it indicates that the elements of the sequence get arbitrarily close to a certain point in the space. The behavior of convergent sequences is essential for understanding the stability and continuity of functions and mappings in these spaces.
Distance Function: A distance function, often referred to as a metric, is a mathematical function that defines a notion of distance between elements in a space. In the context of inner product spaces, it measures how far apart two vectors are from each other, based on the properties of the inner product. This concept is essential for understanding various geometric and analytical properties of vector spaces, including convergence and continuity.
Euclidean norm: The Euclidean norm, often represented as ||x||, is a measure of the length or magnitude of a vector in Euclidean space. It is calculated as the square root of the sum of the squares of its components, providing a way to quantify the distance from the origin to the point represented by the vector. This norm is a specific case of a more general concept called norms in inner product spaces, which are fundamental in understanding vector spaces and their properties.
Hausdorff Distance: Hausdorff distance is a measure of how far two subsets of a metric space are from each other. It takes into account the greatest distance from a point in one set to the closest point in the other set, providing a way to quantify the extent of separation between the two sets. This concept is particularly relevant in the context of normed and inner product spaces, as it relates to the idea of distance and how we can characterize different geometrical shapes and their proximity.
Hilbert Space: A Hilbert space is a complete inner product space that provides a generalization of the notion of Euclidean space to infinite dimensions. It is characterized by the presence of an inner product that allows for the measurement of angles and lengths, which enables concepts such as orthogonality and convergence to be defined within this abstract framework.
Inner product: An inner product is a mathematical operation that takes two vectors in an inner product space and produces a scalar, reflecting a notion of angle and length between the vectors. This operation not only defines how to measure lengths and angles but also helps to determine orthogonality, allowing for important properties like linearity and symmetry. It plays a vital role in defining norms and distances, as well as in processes that require orthogonalization of vectors.
Manhattan Distance: Manhattan distance is a metric used to measure the distance between two points in a grid-based path, calculated as the sum of the absolute differences of their Cartesian coordinates. This concept is particularly relevant in contexts where movement can only occur along orthogonal paths, resembling the layout of streets in a city, hence the name 'Manhattan'. It serves as a specific example of a distance metric that helps to analyze spatial relationships and is essential when discussing norms and distances in inner product spaces.
Norm: A norm is a function that assigns a non-negative length or size to vectors in a vector space, capturing the idea of distance and magnitude. Norms are crucial because they allow us to measure how far apart vectors are and how long they are, enabling us to define concepts like convergence and continuity. In inner product spaces, norms are derived from inner products, helping to establish relationships between angles and lengths of vectors.
Orthogonal Complement: The orthogonal complement of a subspace is the set of all vectors in the space that are orthogonal to every vector in that subspace. This concept is vital for understanding the structure of inner product spaces and highlights how different subspaces relate to each other, particularly in terms of distance and projection.
P-norm: The p-norm is a mathematical function that measures the size or length of a vector in a normed vector space, defined as the $p$-th root of the sum of the absolute values of its components raised to the power of $p$. This concept is essential for understanding distances and properties in inner product spaces, where different values of $p$ yield different types of norms, such as the 1-norm (Manhattan distance) and 2-norm (Euclidean distance). The p-norm helps to establish various properties of convergence, continuity, and completeness within these spaces.
Parallelogram Law: The parallelogram law states that in an inner product space, the squared norm of the sum of two vectors is equal to the sum of the squared norms of each vector plus twice the inner product of the two vectors. This law highlights a fundamental relationship between vectors in such spaces, revealing how their lengths and angles interrelate. It is crucial for understanding distances and angles in inner product spaces, linking geometric interpretations with algebraic properties.
Positive Definiteness: Positive definiteness refers to a property of a quadratic form or a symmetric matrix where the associated inner product produces strictly positive values for all non-zero vectors. This concept is crucial because it ensures that the geometric interpretation of inner products is meaningful, leading to unique norms and distances in the vector space, which are vital for understanding the structure of these spaces.
Triangle Inequality: The triangle inequality states that in any inner product space, the distance between two points (or vectors) is always less than or equal to the sum of the distances of each point to a third point. This fundamental property illustrates how the geometry of inner product spaces works, ensuring that the direct path between two points is the shortest. It connects deeply with the concepts of norms and distances, reinforcing how we measure lengths and relationships within these mathematical structures.
Vector projection: Vector projection is the process of projecting one vector onto another, yielding a new vector that represents the component of the first vector in the direction of the second. This concept is crucial in understanding how vectors relate to each other in space and is essential for analyzing angles and distances within inner product spaces. It helps in breaking down complex vector relationships into simpler, manageable parts.