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**Statement List for Metamath Proof Explorer -** 8901-9000 - Page 90 of 107
Type	Label	Description
Statement

Theorem	his2sub2t 8901	Distributive law for inner product of vector subtraction.
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Theorem	hiret 8902	A necessary and sufficient condition for an inner product to be real.
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Theorem	hiidrclt 8903	Real closure of inner product with self.


Theorem	hi01t 8904	Inner product with the 0 vector.


Theorem	hi02t 8905	Inner product with the 0 vector.


Theorem	hiidge0t 8906	Inner product with self is not negative.


Theorem	his6t 8907	Zero inner product with self means vector is zero. Lemma 3.1(S6) of [Beran] p. 95.


Theorem	his1 8908	Conjugate law for inner product. Postulate (S1) of [Beran] p. 95.


Theorem	abshicomt 8909	Commuted inner products have the same absolute values.
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Theorem	hial0 8910	A vector whose inner product is always zero is zero.


Theorem	hial02 8911	A vector whose inner product is always zero is zero.


Theorem	hisubcom 8912	Two vector subtractions simultaneously commute in an inner product.


Theorem	hi2eqt 8913	Lemma used to prove equality of vectors.
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Theorem	hial2eqt 8914	Two vectors whose inner product is always equal are equal.
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Theorem	hial2eq2t 8915	Two vectors whose inner product is always equal are equal.
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Theorem	orthcom 8916	Orthogonality commutes.
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Theorem	normlem0 8917	Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97.


Theorem	normlem1 8918	Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97.


Theorem	normlem2 8919	Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97.


Theorem	normlem3 8920	Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97.


Theorem	normlem4 8921	Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97.


Theorem	normlem5 8922	Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97.


Theorem	normlem6 8923	Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97.


Theorem	normlem7 8924	Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97.


Theorem	normlem8 8925	Lemma used to derive properties of norm.


Theorem	normlem9 8926	Lemma used to derive properties of norm.


Theorem	normlem7tALT 8927	Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97.
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Theorem	bcseq 8928	Equality case of Bunjakovaskij-Cauchy-Schwarz inequality. Specifically, in the equality case the two vectors are collinear. Compare bcsHIL 8989.


Theorem	normlem9at 8929	Lemma used to derive properties of norm. Part of Remark 3.4(B) of [Beran] p. 98.
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Norms

Theorem	dfhnorm2 8930	Alternate definition of the norm of a vector of Hilbert space. Definition of norm in [Beran] p. 96.
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Theorem	normf 8931	The norm function maps from Hilbert space to reals.


Theorem	normvalt 8932	The value of the norm of a vector in Hilbert space. Definition of norm in [Beran] p. 96. In the literature, the norm of is usually written as "\|\| \|\|", but we use function value notation to take advantage of our existing theorems about functions.


Theorem	normclt 8933	Real closure of the norm of a vector.


Theorem	normge0t 8934	The norm of a vector is nonnegative.


Theorem	normgt0tOLD 8935	The norm of non-zero vector is positive.