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Type | Label | Description |
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Statement | ||
Theorem | isphl 21401* | The predicate "is a generalized pre-Hilbert (inner product) space". (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΉ = (Scalarβπ) & β’ , = (Β·πβπ) & β’ 0 = (0gβπ) & β’ β = (*πβπΉ) & β’ π = (0gβπΉ) β β’ (π β PreHil β (π β LVec β§ πΉ β *-Ring β§ βπ₯ β π ((π¦ β π β¦ (π¦ , π₯)) β (π LMHom (ringLModβπΉ)) β§ ((π₯ , π₯) = π β π₯ = 0 ) β§ βπ¦ β π ( β β(π₯ , π¦)) = (π¦ , π₯)))) | ||
Theorem | phllvec 21402 | A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.) |
β’ (π β PreHil β π β LVec) | ||
Theorem | phllmod 21403 | A pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.) |
β’ (π β PreHil β π β LMod) | ||
Theorem | phlsrng 21404 | The scalar ring of a pre-Hilbert space is a star ring. (Contributed by Mario Carneiro, 7-Oct-2015.) |
β’ πΉ = (Scalarβπ) β β’ (π β PreHil β πΉ β *-Ring) | ||
Theorem | phllmhm 21405* | The inner product of a pre-Hilbert space is linear in its left argument. (Contributed by Mario Carneiro, 7-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ , = (Β·πβπ) & β’ π = (Baseβπ) & β’ πΊ = (π₯ β π β¦ (π₯ , π΄)) β β’ ((π β PreHil β§ π΄ β π) β πΊ β (π LMHom (ringLModβπΉ))) | ||
Theorem | ipcl 21406 | Closure of the inner product operation in a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ , = (Β·πβπ) & β’ π = (Baseβπ) & β’ πΎ = (BaseβπΉ) β β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β (π΄ , π΅) β πΎ) | ||
Theorem | ipcj 21407 | Conjugate of an inner product in a pre-Hilbert space. Equation I1 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ , = (Β·πβπ) & β’ π = (Baseβπ) & β’ β = (*πβπΉ) β β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β ( β β(π΄ , π΅)) = (π΅ , π΄)) | ||
Theorem | iporthcom 21408 | Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ , = (Β·πβπ) & β’ π = (Baseβπ) & β’ π = (0gβπΉ) β β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β ((π΄ , π΅) = π β (π΅ , π΄) = π)) | ||
Theorem | ip0l 21409 | Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. (Contributed by NM, 5-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ , = (Β·πβπ) & β’ π = (Baseβπ) & β’ π = (0gβπΉ) & β’ 0 = (0gβπ) β β’ ((π β PreHil β§ π΄ β π) β ( 0 , π΄) = π) | ||
Theorem | ip0r 21410 | Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ , = (Β·πβπ) & β’ π = (Baseβπ) & β’ π = (0gβπΉ) & β’ 0 = (0gβπ) β β’ ((π β PreHil β§ π΄ β π) β (π΄ , 0 ) = π) | ||
Theorem | ipeq0 21411 | The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ , = (Β·πβπ) & β’ π = (Baseβπ) & β’ π = (0gβπΉ) & β’ 0 = (0gβπ) β β’ ((π β PreHil β§ π΄ β π) β ((π΄ , π΄) = π β π΄ = 0 )) | ||
Theorem | ipdir 21412 | Distributive law for inner product (right-distributivity). Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ , = (Β·πβπ) & β’ π = (Baseβπ) & β’ + = (+gβπ) & ⒠⨣ = (+gβπΉ) β β’ ((π β PreHil β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄ + π΅) , πΆ) = ((π΄ , πΆ) ⨣ (π΅ , πΆ))) | ||
Theorem | ipdi 21413 | Distributive law for inner product (left-distributivity). (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ , = (Β·πβπ) & β’ π = (Baseβπ) & β’ + = (+gβπ) & ⒠⨣ = (+gβπΉ) β β’ ((π β PreHil β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄ , (π΅ + πΆ)) = ((π΄ , π΅) ⨣ (π΄ , πΆ))) | ||
Theorem | ip2di 21414 | Distributive law for inner product. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ , = (Β·πβπ) & β’ π = (Baseβπ) & β’ + = (+gβπ) & ⒠⨣ = (+gβπΉ) & β’ (π β π β PreHil) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) β β’ (π β ((π΄ + π΅) , (πΆ + π·)) = (((π΄ , πΆ) ⨣ (π΅ , π·)) ⨣ ((π΄ , π·) ⨣ (π΅ , πΆ)))) | ||
Theorem | ipsubdir 21415 | Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ , = (Β·πβπ) & β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ π = (-gβπΉ) β β’ ((π β PreHil β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄ β π΅) , πΆ) = ((π΄ , πΆ)π(π΅ , πΆ))) | ||
Theorem | ipsubdi 21416 | Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ , = (Β·πβπ) & β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ π = (-gβπΉ) β β’ ((π β PreHil β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β (π΄ , (π΅ β πΆ)) = ((π΄ , π΅)π(π΄ , πΆ))) | ||
Theorem | ip2subdi 21417 | Distributive law for inner product subtraction. (Contributed by Mario Carneiro, 8-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ , = (Β·πβπ) & β’ π = (Baseβπ) & β’ β = (-gβπ) & β’ π = (-gβπΉ) & β’ + = (+gβπΉ) & β’ (π β π β PreHil) & β’ (π β π΄ β π) & β’ (π β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π· β π) β β’ (π β ((π΄ β π΅) , (πΆ β π·)) = (((π΄ , πΆ) + (π΅ , π·))π((π΄ , π·) + (π΅ , πΆ)))) | ||
Theorem | ipass 21418 | Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ , = (Β·πβπ) & β’ π = (Baseβπ) & β’ πΎ = (BaseβπΉ) & β’ Β· = ( Β·π βπ) & β’ Γ = (.rβπΉ) β β’ ((π β PreHil β§ (π΄ β πΎ β§ π΅ β π β§ πΆ β π)) β ((π΄ Β· π΅) , πΆ) = (π΄ Γ (π΅ , πΆ))) | ||
Theorem | ipassr 21419 | "Associative" law for second argument of inner product (compare ipass 21418). (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ , = (Β·πβπ) & β’ π = (Baseβπ) & β’ πΎ = (BaseβπΉ) & β’ Β· = ( Β·π βπ) & β’ Γ = (.rβπΉ) & β’ β = (*πβπΉ) β β’ ((π β PreHil β§ (π΄ β π β§ π΅ β π β§ πΆ β πΎ)) β (π΄ , (πΆ Β· π΅)) = ((π΄ , π΅) Γ ( β βπΆ))) | ||
Theorem | ipassr2 21420 | "Associative" law for inner product. Conjugate version of ipassr 21419. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
β’ πΉ = (Scalarβπ) & β’ , = (Β·πβπ) & β’ π = (Baseβπ) & β’ πΎ = (BaseβπΉ) & β’ Β· = ( Β·π βπ) & β’ Γ = (.rβπΉ) & β’ β = (*πβπΉ) β β’ ((π β PreHil β§ (π΄ β π β§ π΅ β π β§ πΆ β πΎ)) β ((π΄ , π΅) Γ πΆ) = (π΄ , (( β βπΆ) Β· π΅))) | ||
Theorem | ipffval 21421* | The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.) |
β’ π = (Baseβπ) & β’ , = (Β·πβπ) & β’ Β· = (Β·ifβπ) β β’ Β· = (π₯ β π, π¦ β π β¦ (π₯ , π¦)) | ||
Theorem | ipfval 21422 | The inner product operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ π = (Baseβπ) & β’ , = (Β·πβπ) & β’ Β· = (Β·ifβπ) β β’ ((π β π β§ π β π) β (π Β· π) = (π , π)) | ||
Theorem | ipfeq 21423 | If the inner product operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ π = (Baseβπ) & β’ , = (Β·πβπ) & β’ Β· = (Β·ifβπ) β β’ ( , Fn (π Γ π) β Β· = , ) | ||
Theorem | ipffn 21424 | The inner product operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.) |
β’ π = (Baseβπ) & β’ , = (Β·ifβπ) β β’ , Fn (π Γ π) | ||
Theorem | phlipf 21425 | The inner product operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.) |
β’ π = (Baseβπ) & β’ , = (Β·ifβπ) & β’ π = (Scalarβπ) & β’ πΎ = (Baseβπ) β β’ (π β PreHil β , :(π Γ π)βΆπΎ) | ||
Theorem | ip2eq 21426* | Two vectors are equal iff their inner products with all other vectors are equal. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.) |
β’ , = (Β·πβπ) & β’ π = (Baseβπ) β β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β (π΄ = π΅ β βπ₯ β π (π₯ , π΄) = (π₯ , π΅))) | ||
Theorem | isphld 21427* | Properties that determine a pre-Hilbert (inner product) space. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 7-Oct-2015.) |
β’ (π β π = (Baseβπ)) & β’ (π β + = (+gβπ)) & β’ (π β Β· = ( Β·π βπ)) & β’ (π β πΌ = (Β·πβπ)) & β’ (π β 0 = (0gβπ)) & β’ (π β πΉ = (Scalarβπ)) & β’ (π β πΎ = (BaseβπΉ)) & β’ (π β ⨣ = (+gβπΉ)) & β’ (π β Γ = (.rβπΉ)) & β’ (π β β = (*πβπΉ)) & β’ (π β π = (0gβπΉ)) & β’ (π β π β LVec) & β’ (π β πΉ β *-Ring) & β’ ((π β§ π₯ β π β§ π¦ β π) β (π₯πΌπ¦) β πΎ) & β’ ((π β§ π β πΎ β§ (π₯ β π β§ π¦ β π β§ π§ β π)) β (((π Β· π₯) + π¦)πΌπ§) = ((π Γ (π₯πΌπ§)) ⨣ (π¦πΌπ§))) & β’ ((π β§ π₯ β π β§ (π₯πΌπ₯) = π) β π₯ = 0 ) & β’ ((π β§ π₯ β π β§ π¦ β π) β ( β β(π₯πΌπ¦)) = (π¦πΌπ₯)) β β’ (π β π β PreHil) | ||
Theorem | phlpropd 21428* | If two structures have the same components (properties), one is a pre-Hilbert space iff the other one is. (Contributed by Mario Carneiro, 8-Oct-2015.) |
β’ (π β π΅ = (BaseβπΎ)) & β’ (π β π΅ = (BaseβπΏ)) & β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(+gβπΎ)π¦) = (π₯(+gβπΏ)π¦)) & β’ (π β πΉ = (ScalarβπΎ)) & β’ (π β πΉ = (ScalarβπΏ)) & β’ π = (BaseβπΉ) & β’ ((π β§ (π₯ β π β§ π¦ β π΅)) β (π₯( Β·π βπΎ)π¦) = (π₯( Β·π βπΏ)π¦)) & β’ ((π β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(Β·πβπΎ)π¦) = (π₯(Β·πβπΏ)π¦)) β β’ (π β (πΎ β PreHil β πΏ β PreHil)) | ||
Theorem | ssipeq 21429 | The inner product on a subspace equals the inner product on the parent space. (Contributed by AV, 19-Oct-2021.) |
β’ π = (π βΎs π) & β’ , = (Β·πβπ) & β’ π = (Β·πβπ) β β’ (π β π β π = , ) | ||
Theorem | phssipval 21430 | The inner product on a subspace in terms of the inner product on the parent space. (Contributed by NM, 28-Jan-2008.) (Revised by AV, 19-Oct-2021.) |
β’ π = (π βΎs π) & β’ , = (Β·πβπ) & β’ π = (Β·πβπ) & β’ π = (LSubSpβπ) β β’ (((π β PreHil β§ π β π) β§ (π΄ β π β§ π΅ β π)) β (π΄ππ΅) = (π΄ , π΅)) | ||
Theorem | phssip 21431 | The inner product (as a function) on a subspace is a restriction of the inner product (as a function) on the parent space. (Contributed by NM, 28-Jan-2008.) (Revised by AV, 19-Oct-2021.) |
β’ π = (π βΎs π) & β’ π = (LSubSpβπ) & β’ Β· = (Β·ifβπ) & β’ π = (Β·ifβπ) β β’ ((π β PreHil β§ π β π) β π = ( Β· βΎ (π Γ π))) | ||
Theorem | phlssphl 21432 | A subspace of an inner product space (pre-Hilbert space) is an inner product space. (Contributed by AV, 25-Sep-2022.) |
β’ π = (π βΎs π) & β’ π = (LSubSpβπ) β β’ ((π β PreHil β§ π β π) β π β PreHil) | ||
Syntax | cocv 21433 | Extend class notation with orthocomplement of a subset. |
class ocv | ||
Syntax | ccss 21434 | Extend class notation with set of closed subspaces. |
class ClSubSp | ||
Syntax | cthl 21435 | Extend class notation with the Hilbert lattice. |
class toHL | ||
Definition | df-ocv 21436* | Define the orthocomplement function in a given set (which usually is a pre-Hilbert space): it associates with a subset its orthogonal subset (which in the case of a closed linear subspace is its orthocomplement). (Contributed by NM, 7-Oct-2011.) |
β’ ocv = (β β V β¦ (π β π« (Baseββ) β¦ {π₯ β (Baseββ) β£ βπ¦ β π (π₯(Β·πββ)π¦) = (0gβ(Scalarββ))})) | ||
Definition | df-css 21437* | Define the set of closed (linear) subspaces of a given pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) |
β’ ClSubSp = (β β V β¦ {π β£ π = ((ocvββ)β((ocvββ)βπ ))}) | ||
Definition | df-thl 21438 | Define the Hilbert lattice of closed subspaces of a given pre-Hilbert space. (Contributed by Mario Carneiro, 25-Oct-2015.) |
β’ toHL = (β β V β¦ ((toIncβ(ClSubSpββ)) sSet β¨(ocβndx), (ocvββ)β©)) | ||
Theorem | ocvfval 21439* | The orthocomplement operation. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ , = (Β·πβπ) & β’ πΉ = (Scalarβπ) & β’ 0 = (0gβπΉ) & β’ β₯ = (ocvβπ) β β’ (π β π β β₯ = (π β π« π β¦ {π₯ β π β£ βπ¦ β π (π₯ , π¦) = 0 })) | ||
Theorem | ocvval 21440* | Value of the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ , = (Β·πβπ) & β’ πΉ = (Scalarβπ) & β’ 0 = (0gβπΉ) & β’ β₯ = (ocvβπ) β β’ (π β π β ( β₯ βπ) = {π₯ β π β£ βπ¦ β π (π₯ , π¦) = 0 }) | ||
Theorem | elocv 21441* | Elementhood in the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ , = (Β·πβπ) & β’ πΉ = (Scalarβπ) & β’ 0 = (0gβπΉ) & β’ β₯ = (ocvβπ) β β’ (π΄ β ( β₯ βπ) β (π β π β§ π΄ β π β§ βπ₯ β π (π΄ , π₯) = 0 )) | ||
Theorem | ocvi 21442 | Property of a member of the orthocomplement of a subset. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ , = (Β·πβπ) & β’ πΉ = (Scalarβπ) & β’ 0 = (0gβπΉ) & β’ β₯ = (ocvβπ) β β’ ((π΄ β ( β₯ βπ) β§ π΅ β π) β (π΄ , π΅) = 0 ) | ||
Theorem | ocvss 21443 | The orthocomplement of a subset is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ β₯ = (ocvβπ) β β’ ( β₯ βπ) β π | ||
Theorem | ocvocv 21444 | A set is contained in its double orthocomplement. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ β₯ = (ocvβπ) β β’ ((π β PreHil β§ π β π) β π β ( β₯ β( β₯ βπ))) | ||
Theorem | ocvlss 21445 | The orthocomplement of a subset is a linear subspace of the pre-Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ β₯ = (ocvβπ) & β’ πΏ = (LSubSpβπ) β β’ ((π β PreHil β§ π β π) β ( β₯ βπ) β πΏ) | ||
Theorem | ocv2ss 21446 | Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ β₯ = (ocvβπ) β β’ (π β π β ( β₯ βπ) β ( β₯ βπ)) | ||
Theorem | ocvin 21447 | An orthocomplement has trivial intersection with the original subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ β₯ = (ocvβπ) & β’ πΏ = (LSubSpβπ) & β’ 0 = (0gβπ) β β’ ((π β PreHil β§ π β πΏ) β (π β© ( β₯ βπ)) = { 0 }) | ||
Theorem | ocvsscon 21448 | Two ways to say that π and π are orthogonal subspaces. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ π = (Baseβπ) & β’ β₯ = (ocvβπ) β β’ ((π β PreHil β§ π β π β§ π β π) β (π β ( β₯ βπ) β π β ( β₯ βπ))) | ||
Theorem | ocvlsp 21449 | The orthocomplement of a linear span. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ π = (Baseβπ) & β’ β₯ = (ocvβπ) & β’ π = (LSpanβπ) β β’ ((π β PreHil β§ π β π) β ( β₯ β(πβπ)) = ( β₯ βπ)) | ||
Theorem | ocv0 21450 | The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ π = (Baseβπ) & β’ β₯ = (ocvβπ) β β’ ( β₯ ββ ) = π | ||
Theorem | ocvz 21451 | The orthocomplement of the zero subspace. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ π = (Baseβπ) & β’ β₯ = (ocvβπ) & β’ 0 = (0gβπ) β β’ (π β PreHil β ( β₯ β{ 0 }) = π) | ||
Theorem | ocv1 21452 | The orthocomplement of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ π = (Baseβπ) & β’ β₯ = (ocvβπ) & β’ 0 = (0gβπ) β β’ (π β PreHil β ( β₯ βπ) = { 0 }) | ||
Theorem | unocv 21453 | The orthocomplement of a union. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ β₯ = (ocvβπ) β β’ ( β₯ β(π΄ βͺ π΅)) = (( β₯ βπ΄) β© ( β₯ βπ΅)) | ||
Theorem | iunocv 21454* | The orthocomplement of an indexed union. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ β₯ = (ocvβπ) & β’ π = (Baseβπ) β β’ ( β₯ ββͺ π₯ β π΄ π΅) = (π β© β© π₯ β π΄ ( β₯ βπ΅)) | ||
Theorem | cssval 21455* | The set of closed subspaces of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.) |
β’ β₯ = (ocvβπ) & β’ πΆ = (ClSubSpβπ) β β’ (π β π β πΆ = {π β£ π = ( β₯ β( β₯ βπ ))}) | ||
Theorem | iscss 21456 | The predicate "is a closed subspace" (of a pre-Hilbert space). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.) |
β’ β₯ = (ocvβπ) & β’ πΆ = (ClSubSpβπ) β β’ (π β π β (π β πΆ β π = ( β₯ β( β₯ βπ)))) | ||
Theorem | cssi 21457 | Property of a closed subspace (of a pre-Hilbert space). (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ β₯ = (ocvβπ) & β’ πΆ = (ClSubSpβπ) β β’ (π β πΆ β π = ( β₯ β( β₯ βπ))) | ||
Theorem | cssss 21458 | A closed subspace is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΆ = (ClSubSpβπ) β β’ (π β πΆ β π β π) | ||
Theorem | iscss2 21459 | It is sufficient to prove that the double orthocomplement is a subset of the target set to show that the set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΆ = (ClSubSpβπ) & β’ β₯ = (ocvβπ) β β’ ((π β PreHil β§ π β π) β (π β πΆ β ( β₯ β( β₯ βπ)) β π)) | ||
Theorem | ocvcss 21460 | The orthocomplement of any set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΆ = (ClSubSpβπ) & β’ β₯ = (ocvβπ) β β’ ((π β PreHil β§ π β π) β ( β₯ βπ) β πΆ) | ||
Theorem | cssincl 21461 | The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ πΆ = (ClSubSpβπ) β β’ ((π β PreHil β§ π΄ β πΆ β§ π΅ β πΆ) β (π΄ β© π΅) β πΆ) | ||
Theorem | css0 21462 | The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ πΆ = (ClSubSpβπ) & β’ 0 = (0gβπ) β β’ (π β PreHil β { 0 } β πΆ) | ||
Theorem | css1 21463 | The whole space is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΆ = (ClSubSpβπ) β β’ (π β PreHil β π β πΆ) | ||
Theorem | csslss 21464 | A closed subspace of a pre-Hilbert space is a linear subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ πΆ = (ClSubSpβπ) & β’ πΏ = (LSubSpβπ) β β’ ((π β PreHil β§ π β πΆ) β π β πΏ) | ||
Theorem | lsmcss 21465 | A subset of a pre-Hilbert space whose double orthocomplement has a projection decomposition is a closed subspace. This is the core of the proof that a topologically closed subspace is algebraically closed in a Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ πΆ = (ClSubSpβπ) & β’ π = (Baseβπ) & β’ β₯ = (ocvβπ) & β’ β = (LSSumβπ) & β’ (π β π β PreHil) & β’ (π β π β π) & β’ (π β ( β₯ β( β₯ βπ)) β (π β ( β₯ βπ))) β β’ (π β π β πΆ) | ||
Theorem | cssmre 21466 | The closed subspaces of a pre-Hilbert space are a Moore system. Unlike many of our other examples of closure systems, this one is not usually an algebraic closure system df-acs 17538: consider the Hilbert space of sequences ββΆβ with convergent sum; the subspace of all sequences with finite support is the classic example of a non-closed subspace, but for every finite set of sequences of finite support, there is a finite-dimensional (and hence closed) subspace containing all of the sequences, so if closed subspaces were an algebraic closure system this would violate acsfiel 17603. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΆ = (ClSubSpβπ) β β’ (π β PreHil β πΆ β (Mooreβπ)) | ||
Theorem | mrccss 21467 | The Moore closure corresponding to the system of closed subspaces is the double orthocomplement operation. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ β₯ = (ocvβπ) & β’ πΆ = (ClSubSpβπ) & β’ πΉ = (mrClsβπΆ) β β’ ((π β PreHil β§ π β π) β (πΉβπ) = ( β₯ β( β₯ βπ))) | ||
Theorem | thlval 21468 | Value of the Hilbert lattice. (Contributed by Mario Carneiro, 25-Oct-2015.) |
β’ πΎ = (toHLβπ) & β’ πΆ = (ClSubSpβπ) & β’ πΌ = (toIncβπΆ) & β’ β₯ = (ocvβπ) β β’ (π β π β πΎ = (πΌ sSet β¨(ocβndx), β₯ β©)) | ||
Theorem | thlbas 21469 | Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof shortened by AV, 11-Nov-2024.) |
β’ πΎ = (toHLβπ) & β’ πΆ = (ClSubSpβπ) β β’ πΆ = (BaseβπΎ) | ||
Theorem | thlbasOLD 21470 | Obsolete proof of thlbas 21469 as of 11-Nov-2024. Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ πΎ = (toHLβπ) & β’ πΆ = (ClSubSpβπ) β β’ πΆ = (BaseβπΎ) | ||
Theorem | thlle 21471 | Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof shortened by AV, 11-Nov-2024.) |
β’ πΎ = (toHLβπ) & β’ πΆ = (ClSubSpβπ) & β’ πΌ = (toIncβπΆ) & β’ β€ = (leβπΌ) β β’ β€ = (leβπΎ) | ||
Theorem | thlleOLD 21472 | Obsolete proof of thlle 21471 as of 11-Nov-2024. Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ πΎ = (toHLβπ) & β’ πΆ = (ClSubSpβπ) & β’ πΌ = (toIncβπΆ) & β’ β€ = (leβπΌ) β β’ β€ = (leβπΎ) | ||
Theorem | thlleval 21473 | Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) |
β’ πΎ = (toHLβπ) & β’ πΆ = (ClSubSpβπ) & β’ β€ = (leβπΎ) β β’ ((π β πΆ β§ π β πΆ) β (π β€ π β π β π)) | ||
Theorem | thloc 21474 | Orthocomplement on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) |
β’ πΎ = (toHLβπ) & β’ β₯ = (ocvβπ) β β’ β₯ = (ocβπΎ) | ||
Syntax | cpj 21475 | Extend class notation with orthogonal projection function. |
class proj | ||
Syntax | chil 21476 | Extend class notation with class of all Hilbert spaces. |
class Hil | ||
Syntax | cobs 21477 | Extend class notation with the set of orthonormal bases. |
class OBasis | ||
Definition | df-pj 21478* | Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 19547, but we restrict the domain of this function to only total projection functions. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ proj = (β β V β¦ ((π₯ β (LSubSpββ) β¦ (π₯(proj1ββ)((ocvββ)βπ₯))) β© (V Γ ((Baseββ) βm (Baseββ))))) | ||
Definition | df-hil 21479 | Define class of all Hilbert spaces. Based on Proposition 4.5, p. 176, Gudrun Kalmbach, Quantum Measures and Spaces, Kluwer, Dordrecht, 1998. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 16-Oct-2015.) |
β’ Hil = {β β PreHil β£ dom (projββ) = (ClSubSpββ)} | ||
Definition | df-obs 21480* | Define the set of all orthonormal bases for a pre-Hilbert space. An orthonormal basis is a set of mutually orthogonal vectors with norm 1 and such that the linear span is dense in the whole space. (As this is an "algebraic" definition, before we have topology available, we express this denseness by saying that the double orthocomplement is the whole space, or equivalently, the single orthocomplement is trivial.) (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ OBasis = (β β PreHil β¦ {π β π« (Baseββ) β£ (βπ₯ β π βπ¦ β π (π₯(Β·πββ)π¦) = if(π₯ = π¦, (1rβ(Scalarββ)), (0gβ(Scalarββ))) β§ ((ocvββ)βπ) = {(0gββ)})}) | ||
Theorem | pjfval 21481* | The value of the projection function. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΏ = (LSubSpβπ) & β’ β₯ = (ocvβπ) & β’ π = (proj1βπ) & β’ πΎ = (projβπ) β β’ πΎ = ((π₯ β πΏ β¦ (π₯π( β₯ βπ₯))) β© (V Γ (π βm π))) | ||
Theorem | pjdm 21482 | A subspace is in the domain of the projection function iff the subspace admits a projection decomposition of the whole space. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΏ = (LSubSpβπ) & β’ β₯ = (ocvβπ) & β’ π = (proj1βπ) & β’ πΎ = (projβπ) β β’ (π β dom πΎ β (π β πΏ β§ (ππ( β₯ βπ)):πβΆπ)) | ||
Theorem | pjpm 21483 | The projection map is a partial function from subspaces of the pre-Hilbert space to total operators. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΏ = (LSubSpβπ) & β’ πΎ = (projβπ) β β’ πΎ β ((π βm π) βpm πΏ) | ||
Theorem | pjfval2 21484* | Value of the projection map with implicit domain. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ β₯ = (ocvβπ) & β’ π = (proj1βπ) & β’ πΎ = (projβπ) β β’ πΎ = (π₯ β dom πΎ β¦ (π₯π( β₯ βπ₯))) | ||
Theorem | pjval 21485 | Value of the projection map. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ β₯ = (ocvβπ) & β’ π = (proj1βπ) & β’ πΎ = (projβπ) β β’ (π β dom πΎ β (πΎβπ) = (ππ( β₯ βπ))) | ||
Theorem | pjdm2 21486 | A subspace is in the domain of the projection function iff the subspace admits a projection decomposition of the whole space. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΏ = (LSubSpβπ) & β’ β₯ = (ocvβπ) & β’ β = (LSSumβπ) & β’ πΎ = (projβπ) β β’ (π β PreHil β (π β dom πΎ β (π β πΏ β§ (π β ( β₯ βπ)) = π))) | ||
Theorem | pjff 21487 | A projection is a linear operator. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ πΎ = (projβπ) β β’ (π β PreHil β πΎ:dom πΎβΆ(π LMHom π)) | ||
Theorem | pjf 21488 | A projection is a function on the base set. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ πΎ = (projβπ) & β’ π = (Baseβπ) β β’ (π β dom πΎ β (πΎβπ):πβΆπ) | ||
Theorem | pjf2 21489 | A projection is a function from the base set to the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ πΎ = (projβπ) & β’ π = (Baseβπ) β β’ ((π β PreHil β§ π β dom πΎ) β (πΎβπ):πβΆπ) | ||
Theorem | pjfo 21490 | A projection is a surjection onto the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ πΎ = (projβπ) & β’ π = (Baseβπ) β β’ ((π β PreHil β§ π β dom πΎ) β (πΎβπ):πβontoβπ) | ||
Theorem | pjcss 21491 | A projection subspace is an (algebraically) closed subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ πΎ = (projβπ) & β’ πΆ = (ClSubSpβπ) β β’ (π β PreHil β dom πΎ β πΆ) | ||
Theorem | ocvpj 21492 | The orthocomplement of a projection subspace is a projection subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ πΎ = (projβπ) & β’ β₯ = (ocvβπ) β β’ ((π β PreHil β§ π β dom πΎ) β ( β₯ βπ) β dom πΎ) | ||
Theorem | ishil 21493 | The predicate "is a Hilbert space" (over a *-division ring). A Hilbert space is a pre-Hilbert space such that all closed subspaces have a projection decomposition. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.) |
β’ πΎ = (projβπ») & β’ πΆ = (ClSubSpβπ») β β’ (π» β Hil β (π» β PreHil β§ dom πΎ = πΆ)) | ||
Theorem | ishil2 21494* | The predicate "is a Hilbert space" (over a *-division ring). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.) |
β’ π = (Baseβπ») & β’ β = (LSSumβπ») & β’ β₯ = (ocvβπ») & β’ πΆ = (ClSubSpβπ») β β’ (π» β Hil β (π» β PreHil β§ βπ β πΆ (π β ( β₯ βπ )) = π)) | ||
Theorem | isobs 21495* | The predicate "is an orthonormal basis" (over a pre-Hilbert space). (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ π = (Baseβπ) & β’ , = (Β·πβπ) & β’ πΉ = (Scalarβπ) & β’ 1 = (1rβπΉ) & β’ 0 = (0gβπΉ) & β’ β₯ = (ocvβπ) & β’ π = (0gβπ) β β’ (π΅ β (OBasisβπ) β (π β PreHil β§ π΅ β π β§ (βπ₯ β π΅ βπ¦ β π΅ (π₯ , π¦) = if(π₯ = π¦, 1 , 0 ) β§ ( β₯ βπ΅) = {π}))) | ||
Theorem | obsip 21496 | The inner product of two elements of an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ π = (Baseβπ) & β’ , = (Β·πβπ) & β’ πΉ = (Scalarβπ) & β’ 1 = (1rβπΉ) & β’ 0 = (0gβπΉ) β β’ ((π΅ β (OBasisβπ) β§ π β π΅ β§ π β π΅) β (π , π) = if(π = π, 1 , 0 )) | ||
Theorem | obsipid 21497 | A basis element has length one. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ , = (Β·πβπ) & β’ πΉ = (Scalarβπ) & β’ 1 = (1rβπΉ) β β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β (π΄ , π΄) = 1 ) | ||
Theorem | obsrcl 21498 | Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ (π΅ β (OBasisβπ) β π β PreHil) | ||
Theorem | obsss 21499 | An orthonormal basis is a subset of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ π = (Baseβπ) β β’ (π΅ β (OBasisβπ) β π΅ β π) | ||
Theorem | obsne0 21500 | A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ 0 = (0gβπ) β β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β π΄ β 0 ) |
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