Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

Showing 41-60 of 123

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Cauchy-Schwarz Inequality for a Positive Semi-Definite Self-Adjoint Operator
lemmalem:positive-semidefinite-cauchy-schwarz-2026bAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$ , and let $T$ be a linear operator on $V$ that is self-adjoint and positive semi-definite. Let $|z|$ denote the modulus of a complex number $z$ . Then the following hold.…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-03T00:34:45.041526Z
A Linear Subspace is a Vector Space and Inherits an Inner Product
lemmalem:subspace-inner-product-space-2026bAnalysis Algebra Linear Algebra
Let $K$ be a field, let $V$ be a vector space over $K$ with zero vector $0_{V}$ , and let $W$ be a linear subspace of $V$ . Then the following hold. 1. (Vector space) The set $W$ , equipped with the restrictions to $W$ of the addition and the scalar multiplication of $V$ , is a vect…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-03T00:34:38.892784Z
The Span of a Finite Family is the Smallest Subspace Containing It
lemmalem:span-is-subspace-2026bAlgebra Linear Algebra
Let $K$ be a field, let $V$ be a vector space over $K$ , let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$ , let $v\in V^{n}$ be an $n$ -tuple in $V$ with components $v_{k}$ , and let $\operatorname{span}(v)$ be its span. Then the following hold.…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-03T00:34:20.469367Z
Finite-Dimensional Vector Space
definitiondef:finite-dimensional-vector-space-2026bAlgebra Linear Algebra
Let $K$ be a field and let $V$ be a vector space over $K$ with zero vector $0_{V}$ . The space $V$ is finite-dimensional if $V=\{0_{V}\}$ , or if there exist a natural number $n$ and an $n$ -tuple $e\in V^{n}$ that is a basis of $V$ .
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-03T00:34:14.833470Z
Span of a Finite Family of Vectors
definitiondef:span-finite-family-2026bAlgebra Linear Algebra
Let $K$ be a field, let $V$ be a vector space over $K$ , let $n$ be a natural number, and let $v\in V^{n}$ be an $n$ -tuple in $V$ , with components $v_{k}$ . The span of $v$ is the set…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-03T00:34:11.526107Z
Coordinate Isometry Determined by a Finite Orthonormal Basis
lemmalem:coordinate-isometry-orthonormal-basis-2026cAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$ and induced norm $\lVert\cdot\rVert$ , and let $d$ be the map with $d(u,v)=\lVert u-v\rVert$ , which is a metric on $V$ by claim 3 of…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-03T00:14:25.141197Z
Self-Adjointness, Unitarity and Orthogonal Projections Through the Adjoint in Finite Dimensions
lemmalem:operator-classes-via-adjoint-2026dAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space that has an orthonormal basis $e\in V^{n}$ for some natural number $n$ , where $V^{n}$ is the set of $n$ -tuples in $V$ . By Uniqueness of the Adjoint, and Existence in Finite Dimensions every…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:57:11.365268Z
Every Linear Operator on a Space with a Finite Orthonormal Basis is Bounded
lemmalem:finite-orthonormal-basis-operator-bounded-2026bAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with induced norm $\lVert\cdot\rVert$ , which is a norm on $V$ by claim 2 of The Induced Norm is a Norm, and Induces a Metric. Let $n$ be a natural number, let $e\in V^{n}$ be an $n$ -tuple in $V$ th…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:55:50.113931Z
Properties of the Operator Norm
lemmalem:operator-norm-properties-2026aAnalysis Linear Algebra
Let $V$ be a complex vector space equipped with a norm $\lVert\cdot\rVert$ , let $S$ and $T$ be bounded linear operators on $V$ , and let $\lambda$ be a complex number with modulus $|\lambda|$ . Write $\lVert S\rVert_{\mathrm{op}}$ and $\lVert T\rVert_{\mathrm{op}}$ for their…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:54:33.169849Z
Operator Norm
definitiondef:operator-norm-2026aAnalysis Linear Algebra
Let $V$ be a complex vector space equipped with a norm $\lVert\cdot\rVert$ , let $T$ be a bounded linear operator on $V$ , and let $c$ be a real number. The number $c$ is an operator norm of $T$ if $c$ is a bound for $T$ and $c\le C\qquad\text{for every bound }C\text{ for }T,$ …
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:53:37.793066Z
Existence and Uniqueness of the Operator Norm
lemmalem:operator-norm-existence-uniqueness-2026aAnalysis Linear Algebra
Let $V$ be a complex vector space equipped with a norm $\lVert\cdot\rVert$ , and let $T$ be a bounded linear operator on $V$ . Let $B_{T}$ denote the set of those real numbers that are of the form $\lVert T(u)\rVert$ for some $u\in V$ with $\lVert u\rVert\le1$ , the order being that…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:53:34.030018Z
Bound for a Linear Operator and Bounded Linear Operator
definitiondef:bounded-linear-operator-2026aAnalysis Linear Algebra
Let $V$ be a complex vector space equipped with a norm $\lVert\cdot\rVert$ , let $T$ be a linear operator on $V$ , and let $C$ be a real number with $0\le C$ , the order being that of the ordered field of real numbers. The number $C$ is a bound for $T$ if…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:52:58.902093Z
Triangle Inequality for Finite Sums of Vectors
lemmalem:finite-sum-norm-triangle-2026aAnalysis Linear Algebra
Let $V$ be a complex vector space equipped with a norm $\lVert\cdot\rVert$ , let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$ , and let $v:[n]\to V$ be a map with values $v_{k}$ . Then…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:52:55.680935Z
Positive Definite Operator
definitiondef:positive-definite-operator-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$ , and let $T$ be a linear operator on $V$ . The operator $T$ is positive definite if for every $u\in V$ the complex number $\langle u,T(u)\rangle$ is a real number satisfyin…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:52:50.972820Z
Properties of Unitary Operators
lemmalem:unitary-preserves-inner-product-2026bAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, with induced norm $\lVert\cdot\rVert$ , which is a norm on $V$ by claim 2 of The Induced Norm is a Norm, and Induces a Metric. Let $T$ be a unitary operator on $V$ . Throughout, $\lVert\cdot\rVert$ …
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:24:07.669932Z
Orthogonal Projection
definitiondef:orthogonal-projection-2026bAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, let $P$ be a linear operator on $V$ , and let the product of operators be as in that definition. The operator $P$ is an orthogonal projection if it is self-adjoint and satisfies $PP=P.$
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:23:28.754068Z
Positive Semi-Definite Operator
definitiondef:positive-semidefinite-operator-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space and let $T$ be a linear operator on $V$ . The operator $T$ is positive semi-definite if for every $u\in V$ the complex number $\langle u,T(u)\rangle$ is a real number satisfying…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:23:24.689519Z
Unitary Operator
definitiondef:unitary-operator-2026bAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space and let $T$ be a linear operator on $V$ . The operator $T$ is unitary if it is surjective, that is, every $v\in V$ satisfies $v=T(u)$ for some $u\in V$ , and if…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:23:21.164486Z
Self-Adjoint Operator
definitiondef:self-adjoint-operator-2026bAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space and let $T$ be a linear operator on $V$ . The operator $T$ is self-adjoint if $\langle T(u),v\rangle=\langle u,T(v)\rangle\qquad\text{for all }u,v\in V.$
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:23:16.975158Z
Algebraic Properties of the Adjoint in Finite Dimensions
lemmalem:adjoint-properties-2026cAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space that has an orthonormal basis $e\in V^{n}$ for some natural number $n$ , where $V^{n}$ is the set of $n$ -tuples in $V$ . By Uniqueness of the Adjoint, and Existence in Finite Dimensions every…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:23:13.713069Z

Showing 41-60 of 123

Back Next

Cauchy-Schwarz Inequality for a Positive Semi-Definite Self-Adjoint Operator

A Linear Subspace is a Vector Space and Inherits an Inner Product

The Span of a Finite Family is the Smallest Subspace Containing It

Finite-Dimensional Vector Space

Span of a Finite Family of Vectors

Coordinate Isometry Determined by a Finite Orthonormal Basis

Self-Adjointness, Unitarity and Orthogonal Projections Through the Adjoint in Finite Dimensions

Every Linear Operator on a Space with a Finite Orthonormal Basis is Bounded

Properties of the Operator Norm

Operator Norm

Existence and Uniqueness of the Operator Norm

Bound for a Linear Operator and Bounded Linear Operator

Triangle Inequality for Finite Sums of Vectors

Positive Definite Operator

Properties of Unitary Operators

Orthogonal Projection

Positive Semi-Definite Operator

Unitary Operator

Self-Adjoint Operator

Algebraic Properties of the Adjoint in Finite Dimensions