Theorems

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

Showing 141-160 of 306

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Concatenation of Finite Sums
lemmalem:finite-sum-concatenation-2026bAnalysis Algebra
Let $K$ be a field. Let $\mathbb{N}$ be the set of natural numbers, with addition $+$ and successor map $S$ as in that definition, and for a natural number $p$ let $[p]$ be the initial segment determined by $p$ , that is, the set of natural numbers $k$ with $1\le k\le p$ . Let…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-03T00:13:48.718428Z
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
Adjoint of a Linear Operator
definitiondef:adjoint-operator-2026bAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space and let $S$ and $T$ be linear operators on $V$ . The operator $S$ is an adjoint of $T$ if $\langle S(u),v\rangle=\langle u,T(v)\rangle\qquad\text{for all }u,v\in V.$
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:22:27.867455Z
Uniqueness of the Adjoint, and Existence in Finite Dimensions
theoremthm:adjoint-existence-uniqueness-2026cAnalysis 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, and let $T$ be a linear operator on $V$ . Adjoints are as in…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:22:23.946910Z
The Standard Basis of the Complex Coordinate Space is an Orthonormal Basis
lemmalem:standard-basis-cn-orthonormal-2026bAnalysis Linear Algebra
Let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$ , and let $\mathbb{C}^{n}$ be the complex coordinate space, which is a complex vector space by The Complex Coordinate Space is a Complex Vector Space and, together with the standard inner product…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:20:09.854836Z
Orthonormal Expansion and Parseval's Identity in Finite Dimensions
theoremthm:orthonormal-expansion-parseval-2026bAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with induced norm $\lVert\cdot\rVert$ , let $n$ be a natural number, and let $e\in V^{n}$ be an $n$ -tuple in $V$ that is orthonormal, with components $e_{k}$ . Sums of vectors are finite sums in $V$ …
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:07:35.182805Z
Orthonormal Basis of a Complex Inner Product Space
definitiondef:orthonormal-basis-2026bAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$ , and let $e\in V^{n}$ be an $n$ -tuple in $V$ , that is, a map from $[n]$ to $V$ , with components $e_{k}$ . The tuple…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-02T20:07:19.382362Z

Showing 141-160 of 306

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Concatenation of Finite Sums

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

Adjoint of a Linear Operator

Uniqueness of the Adjoint, and Existence in Finite Dimensions

The Standard Basis of the Complex Coordinate Space is an Orthonormal Basis

Orthonormal Expansion and Parseval's Identity in Finite Dimensions

Orthonormal Basis of a Complex Inner Product Space