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Theorem List for Metamath Proof Explorer - 30301-30400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhv2times 30301 Two times a vector. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (2 Β·β„Ž 𝐴) = (𝐴 +β„Ž 𝐴))
 
Theoremhvnegdii 30302 Distribution of negative over subtraction. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (-1 Β·β„Ž (𝐴 βˆ’β„Ž 𝐡)) = (𝐡 βˆ’β„Ž 𝐴)
 
Theoremhvsubeq0i 30303 If the difference between two vectors is zero, they are equal. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   ((𝐴 βˆ’β„Ž 𝐡) = 0β„Ž ↔ 𝐴 = 𝐡)
 
Theoremhvsubcan2i 30304 Vector cancellation law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   ((𝐴 +β„Ž 𝐡) +β„Ž (𝐴 βˆ’β„Ž 𝐡)) = (2 Β·β„Ž 𝐴)
 
Theoremhvaddcani 30305 Cancellation law for vector addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   ((𝐴 +β„Ž 𝐡) = (𝐴 +β„Ž 𝐢) ↔ 𝐡 = 𝐢)
 
Theoremhvsubaddi 30306 Relationship between vector subtraction and addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   ((𝐴 βˆ’β„Ž 𝐡) = 𝐢 ↔ (𝐡 +β„Ž 𝐢) = 𝐴)
 
Theoremhvnegdi 30307 Distribution of negative over subtraction. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (-1 Β·β„Ž (𝐴 βˆ’β„Ž 𝐡)) = (𝐡 βˆ’β„Ž 𝐴))
 
Theoremhvsubeq0 30308 If the difference between two vectors is zero, they are equal. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) = 0β„Ž ↔ 𝐴 = 𝐡))
 
Theoremhvaddeq0 30309 If the sum of two vectors is zero, one is the negative of the other. (Contributed by NM, 10-Jun-2006.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) = 0β„Ž ↔ 𝐴 = (-1 Β·β„Ž 𝐡)))
 
Theoremhvaddcan 30310 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) = (𝐴 +β„Ž 𝐢) ↔ 𝐡 = 𝐢))
 
Theoremhvaddcan2 30311 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐢) = (𝐡 +β„Ž 𝐢) ↔ 𝐴 = 𝐡))
 
Theoremhvmulcan 30312 Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
(((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0) ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 Β·β„Ž 𝐡) = (𝐴 Β·β„Ž 𝐢) ↔ 𝐡 = 𝐢))
 
Theoremhvmulcan2 30313 Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ (𝐢 ∈ β„‹ ∧ 𝐢 β‰  0β„Ž)) β†’ ((𝐴 Β·β„Ž 𝐢) = (𝐡 Β·β„Ž 𝐢) ↔ 𝐴 = 𝐡))
 
Theoremhvsubcan 30314 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) = (𝐴 βˆ’β„Ž 𝐢) ↔ 𝐡 = 𝐢))
 
Theoremhvsubcan2 30315 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐢) = (𝐡 βˆ’β„Ž 𝐢) ↔ 𝐴 = 𝐡))
 
Theoremhvsub0 30316 Subtraction of a zero vector. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (𝐴 βˆ’β„Ž 0β„Ž) = 𝐴)
 
Theoremhvsubadd 30317 Relationship between vector subtraction and addition. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) = 𝐢 ↔ (𝐡 +β„Ž 𝐢) = 𝐴))
 
Theoremhvaddsub4 30318 Hilbert vector space addition/subtraction law. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
(((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ ((𝐴 +β„Ž 𝐡) = (𝐢 +β„Ž 𝐷) ↔ (𝐴 βˆ’β„Ž 𝐢) = (𝐷 βˆ’β„Ž 𝐡)))
 
20.1.6  Inner product postulates for a Hilbert space
 
Axiomax-hfi 30319 Inner product maps pairs from β„‹ to β„‚. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
Β·ih :( β„‹ Γ— β„‹)βŸΆβ„‚
 
Theoremhicl 30320 Closure of inner product. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 Β·ih 𝐡) ∈ β„‚)
 
Theoremhicli 30321 Closure inference for inner product. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (𝐴 Β·ih 𝐡) ∈ β„‚
 
Axiomax-his1 30322 Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that βˆ—β€˜π‘₯ is the complex conjugate cjval 15045 of π‘₯. In the literature, the inner product of 𝐴 and 𝐡 is usually written ⟨𝐴, 𝐡⟩, but our operation notation co 7405 allows to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 4634. Physicists use ⟨𝐡 ∣ 𝐴⟩, called Dirac bra-ket notation, to represent this operation; see comments in df-bra 31090. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 Β·ih 𝐡) = (βˆ—β€˜(𝐡 Β·ih 𝐴)))
 
Axiomax-his2 30323 Distributive law for inner product. Postulate (S2) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 +β„Ž 𝐡) Β·ih 𝐢) = ((𝐴 Β·ih 𝐢) + (𝐡 Β·ih 𝐢)))
 
Axiomax-his3 30324 Associative law for inner product. Postulate (S3) of [Beran] p. 95. Warning: Mathematics textbooks usually use our version of the axiom. Physics textbooks, on the other hand, usually replace the left-hand side with (𝐡 Β·ih (𝐴 Β·β„Ž 𝐢)) (e.g., Equation 1.21b of [Hughes] p. 44; Definition (iii) of [ReedSimon] p. 36). See the comments in df-bra 31090 for why the physics definition is swapped. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 Β·β„Ž 𝐡) Β·ih 𝐢) = (𝐴 Β· (𝐡 Β·ih 𝐢)))
 
Axiomax-his4 30325 Identity law for inner product. Postulate (S4) of [Beran] p. 95. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 0 < (𝐴 Β·ih 𝐴))
 
20.2  Inner product and norms
 
20.2.1  Inner product
 
Theoremhis5 30326 Associative law for inner product. Lemma 3.1(S5) of [Beran] p. 95. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐡 Β·ih (𝐴 Β·β„Ž 𝐢)) = ((βˆ—β€˜π΄) Β· (𝐡 Β·ih 𝐢)))
 
Theoremhis52 30327 Associative law for inner product. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐡 Β·ih ((βˆ—β€˜π΄) Β·β„Ž 𝐢)) = (𝐴 Β· (𝐡 Β·ih 𝐢)))
 
Theoremhis35 30328 Move scalar multiplication to outside of inner product. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
(((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ ((𝐴 Β·β„Ž 𝐢) Β·ih (𝐡 Β·β„Ž 𝐷)) = ((𝐴 Β· (βˆ—β€˜π΅)) Β· (𝐢 Β·ih 𝐷)))
 
Theoremhis35i 30329 Move scalar multiplication to outside of inner product. (Contributed by NM, 1-Jul-2005.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‚    &   πΆ ∈ β„‹    &   π· ∈ β„‹    β‡’   ((𝐴 Β·β„Ž 𝐢) Β·ih (𝐡 Β·β„Ž 𝐷)) = ((𝐴 Β· (βˆ—β€˜π΅)) Β· (𝐢 Β·ih 𝐷))
 
Theoremhis7 30330 Distributive law for inner product. Lemma 3.1(S7) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 Β·ih (𝐡 +β„Ž 𝐢)) = ((𝐴 Β·ih 𝐡) + (𝐴 Β·ih 𝐢)))
 
Theoremhiassdi 30331 Distributive/associative law for inner product, useful for linearity proofs. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)
(((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ (((𝐴 Β·β„Ž 𝐡) +β„Ž 𝐢) Β·ih 𝐷) = ((𝐴 Β· (𝐡 Β·ih 𝐷)) + (𝐢 Β·ih 𝐷)))
 
Theoremhis2sub 30332 Distributive law for inner product of vector subtraction. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) Β·ih 𝐢) = ((𝐴 Β·ih 𝐢) βˆ’ (𝐡 Β·ih 𝐢)))
 
Theoremhis2sub2 30333 Distributive law for inner product of vector subtraction. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐴 Β·ih (𝐡 βˆ’β„Ž 𝐢)) = ((𝐴 Β·ih 𝐡) βˆ’ (𝐴 Β·ih 𝐢)))
 
Theoremhire 30334 A necessary and sufficient condition for an inner product to be real. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 Β·ih 𝐡) ∈ ℝ ↔ (𝐴 Β·ih 𝐡) = (𝐡 Β·ih 𝐴)))
 
Theoremhiidrcl 30335 Real closure of inner product with self. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (𝐴 Β·ih 𝐴) ∈ ℝ)
 
Theoremhi01 30336 Inner product with the 0 vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (0β„Ž Β·ih 𝐴) = 0)
 
Theoremhi02 30337 Inner product with the 0 vector. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (𝐴 Β·ih 0β„Ž) = 0)
 
Theoremhiidge0 30338 Inner product with self is not negative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ 0 ≀ (𝐴 Β·ih 𝐴))
 
Theoremhis6 30339 Zero inner product with self means vector is zero. Lemma 3.1(S6) of [Beran] p. 95. (Contributed by NM, 27-Jul-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ ((𝐴 Β·ih 𝐴) = 0 ↔ 𝐴 = 0β„Ž))
 
Theoremhis1i 30340 Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. (Contributed by NM, 15-May-2005.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (𝐴 Β·ih 𝐡) = (βˆ—β€˜(𝐡 Β·ih 𝐴))
 
Theoremabshicom 30341 Commuted inner products have the same absolute values. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (absβ€˜(𝐴 Β·ih 𝐡)) = (absβ€˜(𝐡 Β·ih 𝐴)))
 
Theoremhial0 30342* A vector whose inner product is always zero is zero. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (βˆ€π‘₯ ∈ β„‹ (𝐴 Β·ih π‘₯) = 0 ↔ 𝐴 = 0β„Ž))
 
Theoremhial02 30343* A vector whose inner product is always zero is zero. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (βˆ€π‘₯ ∈ β„‹ (π‘₯ Β·ih 𝐴) = 0 ↔ 𝐴 = 0β„Ž))
 
Theoremhisubcomi 30344 Two vector subtractions simultaneously commute in an inner product. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    &   π· ∈ β„‹    β‡’   ((𝐴 βˆ’β„Ž 𝐡) Β·ih (𝐢 βˆ’β„Ž 𝐷)) = ((𝐡 βˆ’β„Ž 𝐴) Β·ih (𝐷 βˆ’β„Ž 𝐢))
 
Theoremhi2eq 30345 Lemma used to prove equality of vectors. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 Β·ih (𝐴 βˆ’β„Ž 𝐡)) = (𝐡 Β·ih (𝐴 βˆ’β„Ž 𝐡)) ↔ 𝐴 = 𝐡))
 
Theoremhial2eq 30346* Two vectors whose inner product is always equal are equal. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (βˆ€π‘₯ ∈ β„‹ (𝐴 Β·ih π‘₯) = (𝐡 Β·ih π‘₯) ↔ 𝐴 = 𝐡))
 
Theoremhial2eq2 30347* Two vectors whose inner product is always equal are equal. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (βˆ€π‘₯ ∈ β„‹ (π‘₯ Β·ih 𝐴) = (π‘₯ Β·ih 𝐡) ↔ 𝐴 = 𝐡))
 
Theoremorthcom 30348 Orthogonality commutes. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 Β·ih 𝐡) = 0 ↔ (𝐡 Β·ih 𝐴) = 0))
 
Theoremnormlem0 30349 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 7-Oct-1999.) (New usage is discouraged.)
𝑆 ∈ β„‚    &   πΉ ∈ β„‹    &   πΊ ∈ β„‹    β‡’   ((𝐹 βˆ’β„Ž (𝑆 Β·β„Ž 𝐺)) Β·ih (𝐹 βˆ’β„Ž (𝑆 Β·β„Ž 𝐺))) = (((𝐹 Β·ih 𝐹) + (-(βˆ—β€˜π‘†) Β· (𝐹 Β·ih 𝐺))) + ((-𝑆 Β· (𝐺 Β·ih 𝐹)) + ((𝑆 Β· (βˆ—β€˜π‘†)) Β· (𝐺 Β·ih 𝐺))))
 
Theoremnormlem1 30350 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 22-Aug-1999.) (New usage is discouraged.)
𝑆 ∈ β„‚    &   πΉ ∈ β„‹    &   πΊ ∈ β„‹    &   π‘… ∈ ℝ    &   (absβ€˜π‘†) = 1    β‡’   ((𝐹 βˆ’β„Ž ((𝑆 Β· 𝑅) Β·β„Ž 𝐺)) Β·ih (𝐹 βˆ’β„Ž ((𝑆 Β· 𝑅) Β·β„Ž 𝐺))) = (((𝐹 Β·ih 𝐹) + (((βˆ—β€˜π‘†) Β· -𝑅) Β· (𝐹 Β·ih 𝐺))) + (((𝑆 Β· -𝑅) Β· (𝐺 Β·ih 𝐹)) + ((𝑅↑2) Β· (𝐺 Β·ih 𝐺))))
 
Theoremnormlem2 30351 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 27-Jul-1999.) (New usage is discouraged.)
𝑆 ∈ β„‚    &   πΉ ∈ β„‹    &   πΊ ∈ β„‹    &   π΅ = -(((βˆ—β€˜π‘†) Β· (𝐹 Β·ih 𝐺)) + (𝑆 Β· (𝐺 Β·ih 𝐹)))    β‡’   π΅ ∈ ℝ
 
Theoremnormlem3 30352 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
𝑆 ∈ β„‚    &   πΉ ∈ β„‹    &   πΊ ∈ β„‹    &   π΅ = -(((βˆ—β€˜π‘†) Β· (𝐹 Β·ih 𝐺)) + (𝑆 Β· (𝐺 Β·ih 𝐹)))    &   π΄ = (𝐺 Β·ih 𝐺)    &   πΆ = (𝐹 Β·ih 𝐹)    &   π‘… ∈ ℝ    β‡’   (((𝐴 Β· (𝑅↑2)) + (𝐡 Β· 𝑅)) + 𝐢) = (((𝐹 Β·ih 𝐹) + (((βˆ—β€˜π‘†) Β· -𝑅) Β· (𝐹 Β·ih 𝐺))) + (((𝑆 Β· -𝑅) Β· (𝐺 Β·ih 𝐹)) + ((𝑅↑2) Β· (𝐺 Β·ih 𝐺))))
 
Theoremnormlem4 30353 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
𝑆 ∈ β„‚    &   πΉ ∈ β„‹    &   πΊ ∈ β„‹    &   π΅ = -(((βˆ—β€˜π‘†) Β· (𝐹 Β·ih 𝐺)) + (𝑆 Β· (𝐺 Β·ih 𝐹)))    &   π΄ = (𝐺 Β·ih 𝐺)    &   πΆ = (𝐹 Β·ih 𝐹)    &   π‘… ∈ ℝ    &   (absβ€˜π‘†) = 1    β‡’   ((𝐹 βˆ’β„Ž ((𝑆 Β· 𝑅) Β·β„Ž 𝐺)) Β·ih (𝐹 βˆ’β„Ž ((𝑆 Β· 𝑅) Β·β„Ž 𝐺))) = (((𝐴 Β· (𝑅↑2)) + (𝐡 Β· 𝑅)) + 𝐢)
 
Theoremnormlem5 30354 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 10-Aug-1999.) (New usage is discouraged.)
𝑆 ∈ β„‚    &   πΉ ∈ β„‹    &   πΊ ∈ β„‹    &   π΅ = -(((βˆ—β€˜π‘†) Β· (𝐹 Β·ih 𝐺)) + (𝑆 Β· (𝐺 Β·ih 𝐹)))    &   π΄ = (𝐺 Β·ih 𝐺)    &   πΆ = (𝐹 Β·ih 𝐹)    &   π‘… ∈ ℝ    &   (absβ€˜π‘†) = 1    β‡’   0 ≀ (((𝐴 Β· (𝑅↑2)) + (𝐡 Β· 𝑅)) + 𝐢)
 
Theoremnormlem6 30355 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 4-Jun-2014.) (New usage is discouraged.)
𝑆 ∈ β„‚    &   πΉ ∈ β„‹    &   πΊ ∈ β„‹    &   π΅ = -(((βˆ—β€˜π‘†) Β· (𝐹 Β·ih 𝐺)) + (𝑆 Β· (𝐺 Β·ih 𝐹)))    &   π΄ = (𝐺 Β·ih 𝐺)    &   πΆ = (𝐹 Β·ih 𝐹)    &   (absβ€˜π‘†) = 1    β‡’   (absβ€˜π΅) ≀ (2 Β· ((βˆšβ€˜π΄) Β· (βˆšβ€˜πΆ)))
 
Theoremnormlem7 30356 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
𝑆 ∈ β„‚    &   πΉ ∈ β„‹    &   πΊ ∈ β„‹    &   (absβ€˜π‘†) = 1    β‡’   (((βˆ—β€˜π‘†) Β· (𝐹 Β·ih 𝐺)) + (𝑆 Β· (𝐺 Β·ih 𝐹))) ≀ (2 Β· ((βˆšβ€˜(𝐺 Β·ih 𝐺)) Β· (βˆšβ€˜(𝐹 Β·ih 𝐹))))
 
Theoremnormlem8 30357 Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    &   π· ∈ β„‹    β‡’   ((𝐴 +β„Ž 𝐡) Β·ih (𝐢 +β„Ž 𝐷)) = (((𝐴 Β·ih 𝐢) + (𝐡 Β·ih 𝐷)) + ((𝐴 Β·ih 𝐷) + (𝐡 Β·ih 𝐢)))
 
Theoremnormlem9 30358 Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    &   π· ∈ β„‹    β‡’   ((𝐴 βˆ’β„Ž 𝐡) Β·ih (𝐢 βˆ’β„Ž 𝐷)) = (((𝐴 Β·ih 𝐢) + (𝐡 Β·ih 𝐷)) βˆ’ ((𝐴 Β·ih 𝐷) + (𝐡 Β·ih 𝐢)))
 
Theoremnormlem7tALT 30359 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   ((𝑆 ∈ β„‚ ∧ (absβ€˜π‘†) = 1) β†’ (((βˆ—β€˜π‘†) Β· (𝐴 Β·ih 𝐡)) + (𝑆 Β· (𝐡 Β·ih 𝐴))) ≀ (2 Β· ((βˆšβ€˜(𝐡 Β·ih 𝐡)) Β· (βˆšβ€˜(𝐴 Β·ih 𝐴)))))
 
Theorembcseqi 30360 Equality case of Bunjakovaskij-Cauchy-Schwarz inequality. Specifically, in the equality case the two vectors are collinear. Compare bcsiHIL 30420. (Contributed by NM, 16-Jul-2001.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (((𝐴 Β·ih 𝐡) Β· (𝐡 Β·ih 𝐴)) = ((𝐴 Β·ih 𝐴) Β· (𝐡 Β·ih 𝐡)) ↔ ((𝐡 Β·ih 𝐡) Β·β„Ž 𝐴) = ((𝐴 Β·ih 𝐡) Β·β„Ž 𝐡))
 
Theoremnormlem9at 30361 Lemma used to derive properties of norm. Part of Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 βˆ’β„Ž 𝐡) Β·ih (𝐴 βˆ’β„Ž 𝐡)) = (((𝐴 Β·ih 𝐴) + (𝐡 Β·ih 𝐡)) βˆ’ ((𝐴 Β·ih 𝐡) + (𝐡 Β·ih 𝐴))))
 
20.2.2  Norms
 
Theoremdfhnorm2 30362 Alternate definition of the norm of a vector of Hilbert space. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
normβ„Ž = (π‘₯ ∈ β„‹ ↦ (βˆšβ€˜(π‘₯ Β·ih π‘₯)))
 
Theoremnormf 30363 The norm function maps from Hilbert space to reals. (Contributed by NM, 6-Sep-2007.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
normβ„Ž: β„‹βŸΆβ„
 
Theoremnormval 30364 The value of the norm of a vector in Hilbert space. Definition of norm in [Beran] p. 96. In the literature, the norm of 𝐴 is usually written as "|| 𝐴 ||", but we use function value notation to take advantage of our existing theorems about functions. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (normβ„Žβ€˜π΄) = (βˆšβ€˜(𝐴 Β·ih 𝐴)))
 
Theoremnormcl 30365 Real closure of the norm of a vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (normβ„Žβ€˜π΄) ∈ ℝ)
 
Theoremnormge0 30366 The norm of a vector is nonnegative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ 0 ≀ (normβ„Žβ€˜π΄))
 
Theoremnormgt0 30367 The norm of nonzero vector is positive. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (𝐴 β‰  0β„Ž ↔ 0 < (normβ„Žβ€˜π΄)))
 
Theoremnorm0 30368 The norm of a zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
(normβ„Žβ€˜0β„Ž) = 0
 
Theoremnorm-i 30369 Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ ((normβ„Žβ€˜π΄) = 0 ↔ 𝐴 = 0β„Ž))
 
Theoremnormne0 30370 A norm is nonzero iff its argument is a nonzero vector. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ ((normβ„Žβ€˜π΄) β‰  0 ↔ 𝐴 β‰  0β„Ž))
 
Theoremnormcli 30371 Real closure of the norm of a vector. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    β‡’   (normβ„Žβ€˜π΄) ∈ ℝ
 
Theoremnormsqi 30372 The square of a norm. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    β‡’   ((normβ„Žβ€˜π΄)↑2) = (𝐴 Β·ih 𝐴)
 
Theoremnorm-i-i 30373 Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 5-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    β‡’   ((normβ„Žβ€˜π΄) = 0 ↔ 𝐴 = 0β„Ž)
 
Theoremnormsq 30374 The square of a norm. (Contributed by NM, 12-May-2005.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ ((normβ„Žβ€˜π΄)↑2) = (𝐴 Β·ih 𝐴))
 
Theoremnormsub0i 30375 Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   ((normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡)) = 0 ↔ 𝐴 = 𝐡)
 
Theoremnormsub0 30376 Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡)) = 0 ↔ 𝐴 = 𝐡))
 
Theoremnorm-ii-i 30377 Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (normβ„Žβ€˜(𝐴 +β„Ž 𝐡)) ≀ ((normβ„Žβ€˜π΄) + (normβ„Žβ€˜π΅))
 
Theoremnorm-ii 30378 Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (normβ„Žβ€˜(𝐴 +β„Ž 𝐡)) ≀ ((normβ„Žβ€˜π΄) + (normβ„Žβ€˜π΅)))
 
Theoremnorm-iii-i 30379 Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
𝐴 ∈ β„‚    &   π΅ ∈ β„‹    β‡’   (normβ„Žβ€˜(𝐴 Β·β„Ž 𝐡)) = ((absβ€˜π΄) Β· (normβ„Žβ€˜π΅))
 
Theoremnorm-iii 30380 Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 25-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‹) β†’ (normβ„Žβ€˜(𝐴 Β·β„Ž 𝐡)) = ((absβ€˜π΄) Β· (normβ„Žβ€˜π΅)))
 
Theoremnormsubi 30381 Negative doesn't change the norm of a Hilbert space vector. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡)) = (normβ„Žβ€˜(𝐡 βˆ’β„Ž 𝐴))
 
Theoremnormpythi 30382 Analogy to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   ((𝐴 Β·ih 𝐡) = 0 β†’ ((normβ„Žβ€˜(𝐴 +β„Ž 𝐡))↑2) = (((normβ„Žβ€˜π΄)↑2) + ((normβ„Žβ€˜π΅)↑2)))
 
Theoremnormsub 30383 Swapping order of subtraction doesn't change the norm of a vector. (Contributed by NM, 14-Aug-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡)) = (normβ„Žβ€˜(𝐡 βˆ’β„Ž 𝐴)))
 
Theoremnormneg 30384 The norm of a vector equals the norm of its negative. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
(𝐴 ∈ β„‹ β†’ (normβ„Žβ€˜(-1 Β·β„Ž 𝐴)) = (normβ„Žβ€˜π΄))
 
Theoremnormpyth 30385 Analogy to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 Β·ih 𝐡) = 0 β†’ ((normβ„Žβ€˜(𝐴 +β„Ž 𝐡))↑2) = (((normβ„Žβ€˜π΄)↑2) + ((normβ„Žβ€˜π΅)↑2))))
 
Theoremnormpyc 30386 Corollary to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 26-Oct-1999.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((𝐴 Β·ih 𝐡) = 0 β†’ (normβ„Žβ€˜π΄) ≀ (normβ„Žβ€˜(𝐴 +β„Ž 𝐡))))
 
Theoremnorm3difi 30387 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   (normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡)) ≀ ((normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐢)) + (normβ„Žβ€˜(𝐢 βˆ’β„Ž 𝐡)))
 
Theoremnorm3adifii 30388 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   (absβ€˜((normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐢)) βˆ’ (normβ„Žβ€˜(𝐡 βˆ’β„Ž 𝐢)))) ≀ (normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡))
 
Theoremnorm3lem 30389 Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    &   π· ∈ ℝ    β‡’   (((normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐢)) < (𝐷 / 2) ∧ (normβ„Žβ€˜(𝐢 βˆ’β„Ž 𝐡)) < (𝐷 / 2)) β†’ (normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡)) < 𝐷)
 
Theoremnorm3dif 30390 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 20-Apr-2006.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡)) ≀ ((normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐢)) + (normβ„Žβ€˜(𝐢 βˆ’β„Ž 𝐡))))
 
Theoremnorm3dif2 30391 Norm of differences around common element. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡)) ≀ ((normβ„Žβ€˜(𝐢 βˆ’β„Ž 𝐴)) + (normβ„Žβ€˜(𝐢 βˆ’β„Ž 𝐡))))
 
Theoremnorm3lemt 30392 Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
(((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ ℝ)) β†’ (((normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐢)) < (𝐷 / 2) ∧ (normβ„Žβ€˜(𝐢 βˆ’β„Ž 𝐡)) < (𝐷 / 2)) β†’ (normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡)) < 𝐷))
 
Theoremnorm3adifi 30393 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 3-Oct-1999.) (New usage is discouraged.)
𝐢 ∈ β„‹    β‡’   ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (absβ€˜((normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐢)) βˆ’ (normβ„Žβ€˜(𝐡 βˆ’β„Ž 𝐢)))) ≀ (normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡)))
 
Theoremnormpari 30394 Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (((normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡))↑2) + ((normβ„Žβ€˜(𝐴 +β„Ž 𝐡))↑2)) = ((2 Β· ((normβ„Žβ€˜π΄)↑2)) + (2 Β· ((normβ„Žβ€˜π΅)↑2)))
 
Theoremnormpar 30395 Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (((normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡))↑2) + ((normβ„Žβ€˜(𝐴 +β„Ž 𝐡))↑2)) = ((2 Β· ((normβ„Žβ€˜π΄)↑2)) + (2 Β· ((normβ„Žβ€˜π΅)↑2))))
 
Theoremnormpar2i 30396 Corollary of parallelogram law for norms. Part of Lemma 3.6 of [Beran] p. 100. (Contributed by NM, 5-Oct-1999.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    β‡’   ((normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡))↑2) = (((2 Β· ((normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐢))↑2)) + (2 Β· ((normβ„Žβ€˜(𝐡 βˆ’β„Ž 𝐢))↑2))) βˆ’ ((normβ„Žβ€˜((𝐴 +β„Ž 𝐡) βˆ’β„Ž (2 Β·β„Ž 𝐢)))↑2))
 
Theorempolid2i 30397 Generalized polarization identity. Generalization of Exercise 4(a) of [ReedSimon] p. 63. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    &   πΆ ∈ β„‹    &   π· ∈ β„‹    β‡’   (𝐴 Β·ih 𝐡) = (((((𝐴 +β„Ž 𝐢) Β·ih (𝐷 +β„Ž 𝐡)) βˆ’ ((𝐴 βˆ’β„Ž 𝐢) Β·ih (𝐷 βˆ’β„Ž 𝐡))) + (i Β· (((𝐴 +β„Ž (i Β·β„Ž 𝐢)) Β·ih (𝐷 +β„Ž (i Β·β„Ž 𝐡))) βˆ’ ((𝐴 βˆ’β„Ž (i Β·β„Ž 𝐢)) Β·ih (𝐷 βˆ’β„Ž (i Β·β„Ž 𝐡)))))) / 4)
 
Theorempolidi 30398 Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of Axiom ax-his3 30324. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
𝐴 ∈ β„‹    &   π΅ ∈ β„‹    β‡’   (𝐴 Β·ih 𝐡) = (((((normβ„Žβ€˜(𝐴 +β„Ž 𝐡))↑2) βˆ’ ((normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡))↑2)) + (i Β· (((normβ„Žβ€˜(𝐴 +β„Ž (i Β·β„Ž 𝐡)))↑2) βˆ’ ((normβ„Žβ€˜(𝐴 βˆ’β„Ž (i Β·β„Ž 𝐡)))↑2)))) / 4)
 
Theorempolid 30399 Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of Axiom ax-his3 30324. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ (𝐴 Β·ih 𝐡) = (((((normβ„Žβ€˜(𝐴 +β„Ž 𝐡))↑2) βˆ’ ((normβ„Žβ€˜(𝐴 βˆ’β„Ž 𝐡))↑2)) + (i Β· (((normβ„Žβ€˜(𝐴 +β„Ž (i Β·β„Ž 𝐡)))↑2) βˆ’ ((normβ„Žβ€˜(𝐴 βˆ’β„Ž (i Β·β„Ž 𝐡)))↑2)))) / 4))
 
20.2.3  Relate Hilbert space to normed complex vector spaces
 
Theoremhilablo 30400 Hilbert space vector addition is an Abelian group operation. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
+β„Ž ∈ AbelOp
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