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Inner Product Spaces
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Parallelogram Law
In an inner product space, the relationship holds. Example: For vectors and in , .
Spectral Theorem
Every compact self-adjoint operator on a Hilbert space has an orthonormal basis of eigenvectors. Example: The matrix represents a self-adjoint operator on with eigenvectors and .
Riesz Representation Theorem
Every continuous linear functional on a Hilbert space can be represented as an inner product with a unique vector in : for all . Example: In , the space of square-summable sequences, the functional is represented by the sequence .
Bessel's Inequality
For any orthonormal sequence in an inner product space, and any vector , the sum of the squares of the inner products is less than or equal to the square of the norm of : . Example: In , the space of square-summable sequences, taking as standard basis vectors and satisfies the inequality.
Projection onto a Subspace
If is a subspace of an inner product space , the projection of a vector onto is the unique vector in such that is orthogonal to every vector in . Example: Projecting onto the span of in gives .
Adjoint Operator
For a linear operator on a Hilbert space , the adjoint is the unique operator such that for all . Example: For the matrix over , the adjoint is .
Completeness
An inner product space is complete if every Cauchy sequence in the space converges to a limit that is also in the space. Example: with the standard inner product is complete and thus a Hilbert space.
Orthogonal Complement
The orthogonal complement of a subspace in an inner product space is the set of all vectors in that are orthogonal to every vector in . Example: In , the orthogonal complement of the - plane is the -axis.
Cauchy-Schwarz Inequality
For all vectors and in an inner product space, , with equality if and only if and are linearly dependent. Example: In , for and , the inequality holds as .
Norm Properties
A norm on a vector space satisfies: 1) Non-negativity: , 2) Definiteness: if and only if , 3) Homogeneity: , 4) Triangle Inequality: . Example: In , the Euclidean norm satisfies these properties.
Gram-Schmidt Process
A method for obtaining an orthonormal set from a linearly independent set. It proceeds by orthogonalizing the vectors with respect to one another and then normalizing them. Example: Starting with and in , the normalized orthogonal vectors would be and .
Inner Product Definition
An inner product on a vector space over (where is or ) is a function that is conjugate symmetric, linear in its first argument, and positive definite. Example: for .
Orthogonality
Two vectors and in an inner product space are orthogonal if . Example: In , the vectors and are orthogonal.
Triangle Inequality
In an inner product space, the norm of the sum of two vectors is less than or equal to the sum of their norms: . Example: In , for vectors and , .
Self-Adjoint Operator
An operator on a Hilbert space is self-adjoint if , meaning it equals its own adjoint. Example: The matrix is self-adjoint on .
Polarization Identity
The inner product can be expressed in terms of the norm: for real vector spaces , and for complex vector spaces . Example: In , can be computed using the identity.
Schwarz Reflection Principle
A principle stating that an analytic function defined on the upper half-plane that takes real values on the real axis can be extended to the lower half-plane by reflection. Example: If is analytic in the upper half-plane and real on the real axis, then extends to the lower half-plane.
Unitary Operator
A linear operator on a Hilbert space is unitary if , where is the identity operator. Example: The matrix is unitary on .
Parseval's Identity
For an orthonormal basis of a Hilbert space and any vector , the sum of the squares of the inner products equals the square of the norm of : . Example: In with the standard basis, for vector , Parseval's identity confirms that the norm squared is equal to the sum of the squares of the modulus of 's components.
Hilbert Space
A Hilbert space is a complete inner product space. These spaces are generalizations of Euclidean spaces to possibly infinite dimensions. Example: , the space of square-integrable functions on the interval , is a Hilbert space.
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