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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | nvmdi 29901 | Distributive law for scalar product over subtraction. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ π = ( Β·π OLD βπ) β β’ ((π β NrmCVec β§ (π΄ β β β§ π΅ β π β§ πΆ β π)) β (π΄π(π΅ππΆ)) = ((π΄ππ΅)π(π΄ππΆ))) | ||
| Theorem | nvnegneg 29902 | Double negative of a vector. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = ( Β·π OLD βπ) β β’ ((π β NrmCVec β§ π΄ β π) β (-1π(-1ππ΄)) = π΄) | ||
| Theorem | nvmul0or 29903 | If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = ( Β·π OLD βπ) & β’ π = (0vecβπ) β β’ ((π β NrmCVec β§ π΄ β β β§ π΅ β π) β ((π΄ππ΅) = π β (π΄ = 0 β¨ π΅ = π))) | ||
| Theorem | nvrinv 29904 | A vector minus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) & β’ π = (0vecβπ) β β’ ((π β NrmCVec β§ π΄ β π) β (π΄πΊ(-1ππ΄)) = π) | ||
| Theorem | nvlinv 29905 | Minus a vector plus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) & β’ π = (0vecβπ) β β’ ((π β NrmCVec β§ π΄ β π) β ((-1ππ΄)πΊπ΄) = π) | ||
| Theorem | nvpncan2 29906 | Cancellation law for vector subtraction. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( βπ£ βπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((π΄πΊπ΅)ππ΄) = π΅) | ||
| Theorem | nvpncan 29907 | Cancellation law for vector subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( βπ£ βπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((π΄πΊπ΅)ππ΅) = π΄) | ||
| Theorem | nvaddsub 29908 | Commutative/associative law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( βπ£ βπ) β β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π β§ πΆ β π)) β ((π΄πΊπ΅)ππΆ) = ((π΄ππΆ)πΊπ΅)) | ||
| Theorem | nvnpcan 29909 | Cancellation law for a normed complex vector space. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( βπ£ βπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((π΄ππ΅)πΊπ΅) = π΄) | ||
| Theorem | nvaddsub4 29910 | Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( βπ£ βπ) β β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π)) β ((π΄πΊπ΅)π(πΆπΊπ·)) = ((π΄ππΆ)πΊ(π΅ππ·))) | ||
| Theorem | nvmeq0 29911 | The difference between two vectors is zero iff they are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ π = (0vecβπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((π΄ππ΅) = π β π΄ = π΅)) | ||
| Theorem | nvmid 29912 | A vector minus itself is the zero vector. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ π = (0vecβπ) β β’ ((π β NrmCVec β§ π΄ β π) β (π΄ππ΄) = π) | ||
| Theorem | nvf 29913 | Mapping for the norm function. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (normCVβπ) β β’ (π β NrmCVec β π:πβΆβ) | ||
| Theorem | nvcl 29914 | The norm of a normed complex vector space is a real number. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (normCVβπ) β β’ ((π β NrmCVec β§ π΄ β π) β (πβπ΄) β β) | ||
| Theorem | nvcli 29915 | The norm of a normed complex vector space is a real number. (Contributed by NM, 20-Apr-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (normCVβπ) & β’ π β NrmCVec & β’ π΄ β π β β’ (πβπ΄) β β | ||
| Theorem | nvs 29916 | Proportionality property of the norm of a scalar product in a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) β β’ ((π β NrmCVec β§ π΄ β β β§ π΅ β π) β (πβ(π΄ππ΅)) = ((absβπ΄) Β· (πβπ΅))) | ||
| Theorem | nvsge0 29917 | The norm of a scalar product with a nonnegative real. (Contributed by NM, 1-Jan-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) β β’ ((π β NrmCVec β§ (π΄ β β β§ 0 β€ π΄) β§ π΅ β π) β (πβ(π΄ππ΅)) = (π΄ Β· (πβπ΅))) | ||
| Theorem | nvm1 29918 | The norm of the negative of a vector. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) β β’ ((π β NrmCVec β§ π΄ β π) β (πβ(-1ππ΄)) = (πβπ΄)) | ||
| Theorem | nvdif 29919 | The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (πβ(π΄πΊ(-1ππ΅))) = (πβ(π΅πΊ(-1ππ΄)))) | ||
| Theorem | nvpi 29920 | The norm of a vector plus the imaginary scalar product of another. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (πβ(π΄πΊ(iππ΅))) = (πβ(π΅πΊ(-iππ΄)))) | ||
| Theorem | nvz0 29921 | The norm of a zero vector is zero. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.) |
| β’ π = (0vecβπ) & β’ π = (normCVβπ) β β’ (π β NrmCVec β (πβπ) = 0) | ||
| Theorem | nvz 29922 | The norm of a vector is zero iff the vector is zero. First part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (0vecβπ) & β’ π = (normCVβπ) β β’ ((π β NrmCVec β§ π΄ β π) β ((πβπ΄) = 0 β π΄ = π)) | ||
| Theorem | nvtri 29923 | Triangle inequality for the norm of a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = (normCVβπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (πβ(π΄πΊπ΅)) β€ ((πβπ΄) + (πβπ΅))) | ||
| Theorem | nvmtri 29924 | Triangle inequality for the norm of a vector difference. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ π = (normCVβπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (πβ(π΄ππ΅)) β€ ((πβπ΄) + (πβπ΅))) | ||
| Theorem | nvabs 29925 | Norm difference property of a normed complex vector space. Problem 3 of [Kreyszig] p. 64. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (absβ((πβπ΄) β (πβπ΅))) β€ (πβ(π΄πΊ(-1ππ΅)))) | ||
| Theorem | nvge0 29926 | The norm of a normed complex vector space is nonnegative. Second part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 28-Nov-2006.) (Proof shortened by AV, 10-Jul-2022.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (normCVβπ) β β’ ((π β NrmCVec β§ π΄ β π) β 0 β€ (πβπ΄)) | ||
| Theorem | nvgt0 29927 | A nonzero norm is positive. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (0vecβπ) & β’ π = (normCVβπ) β β’ ((π β NrmCVec β§ π΄ β π) β (π΄ β π β 0 < (πβπ΄))) | ||
| Theorem | nv1 29928 | From any nonzero vector, construct a vector whose norm is one. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = ( Β·π OLD βπ) & β’ π = (0vecβπ) & β’ π = (normCVβπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΄ β π) β (πβ((1 / (πβπ΄))ππ΄)) = 1) | ||
| Theorem | nvop 29929 | A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.) |
| β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) β β’ (π β NrmCVec β π = β¨β¨πΊ, πβ©, πβ©) | ||
| Theorem | cnnv 29930 | The set of complex numbers is a normed complex vector space. The vector operation is +, the scalar product is Β·, and the norm function is abs. (Contributed by Steve Rodriguez, 3-Dec-2006.) (New usage is discouraged.) |
| β’ π = β¨β¨ + , Β· β©, absβ© β β’ π β NrmCVec | ||
| Theorem | cnnvg 29931 | The vector addition (group) operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.) |
| β’ π = β¨β¨ + , Β· β©, absβ© β β’ + = ( +π£ βπ) | ||
| Theorem | cnnvba 29932 | The base set of the normed complex vector space of complex numbers. (Contributed by NM, 7-Nov-2007.) (New usage is discouraged.) |
| β’ π = β¨β¨ + , Β· β©, absβ© β β’ β = (BaseSetβπ) | ||
| Theorem | cnnvs 29933 | The scalar product operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.) |
| β’ π = β¨β¨ + , Β· β©, absβ© β β’ Β· = ( Β·π OLD βπ) | ||
| Theorem | cnnvnm 29934 | The norm operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.) |
| β’ π = β¨β¨ + , Β· β©, absβ© β β’ abs = (normCVβπ) | ||
| Theorem | cnnvm 29935 | The vector subtraction operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| β’ π = β¨β¨ + , Β· β©, absβ© β β’ β = ( βπ£ βπ) | ||
| Theorem | elimnv 29936 | Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (0vecβπ) & β’ π β NrmCVec β β’ if(π΄ β π, π΄, π) β π | ||
| Theorem | elimnvu 29937 | Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.) |
| β’ if(π β NrmCVec, π, β¨β¨ + , Β· β©, absβ©) β NrmCVec | ||
| Theorem | imsval 29938 | Value of the induced metric of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| β’ π = ( βπ£ βπ) & β’ π = (normCVβπ) & β’ π· = (IndMetβπ) β β’ (π β NrmCVec β π· = (π β π)) | ||
| Theorem | imsdval 29939 | Value of the induced metric (distance function) of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ π = (normCVβπ) & β’ π· = (IndMetβπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = (πβ(π΄ππ΅))) | ||
| Theorem | imsdval2 29940 | Value of the distance function of the induced metric of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) & β’ π· = (IndMetβπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = (πβ(π΄πΊ(-1ππ΅)))) | ||
| Theorem | nvnd 29941 | The norm of a normed complex vector space expressed in terms of the distance function of its induced metric. Problem 1 of [Kreyszig] p. 63. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (0vecβπ) & β’ π = (normCVβπ) & β’ π· = (IndMetβπ) β β’ ((π β NrmCVec β§ π΄ β π) β (πβπ΄) = (π΄π·π)) | ||
| Theorem | imsdf 29942 | Mapping for the induced metric distance function of a normed complex vector space. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π· = (IndMetβπ) β β’ (π β NrmCVec β π·:(π Γ π)βΆβ) | ||
| Theorem | imsmetlem 29943 | Lemma for imsmet 29944. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = (invβπΊ) & β’ π = ( Β·π OLD βπ) & β’ π = (0vecβπ) & β’ π = (normCVβπ) & β’ π· = (IndMetβπ) & β’ π β NrmCVec β β’ π· β (Metβπ) | ||
| Theorem | imsmet 29944 | The induced metric of a normed complex vector space is a metric space. Part of Definition 2.2-1 of [Kreyszig] p. 58. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π· = (IndMetβπ) β β’ (π β NrmCVec β π· β (Metβπ)) | ||
| Theorem | imsxmet 29945 | The induced metric of a normed complex vector space is an extended metric space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π· = (IndMetβπ) β β’ (π β NrmCVec β π· β (βMetβπ)) | ||
| Theorem | cnims 29946 | The metric induced on the complex numbers. cnmet 24288 proves that it is a metric. (Contributed by Steve Rodriguez, 5-Dec-2006.) (Revised by NM, 15-Jan-2008.) (New usage is discouraged.) |
| β’ π = β¨β¨ + , Β· β©, absβ© & β’ π· = (abs β β ) β β’ π· = (IndMetβπ) | ||
| Theorem | vacn 29947 | Vector addition is jointly continuous in both arguments. (Contributed by Jeff Hankins, 16-Jun-2009.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
| β’ πΆ = (IndMetβπ) & β’ π½ = (MetOpenβπΆ) & β’ πΊ = ( +π£ βπ) β β’ (π β NrmCVec β πΊ β ((π½ Γt π½) Cn π½)) | ||
| Theorem | nmcvcn 29948 | The norm of a normed complex vector space is a continuous function. (Contributed by NM, 16-May-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.) |
| β’ π = (normCVβπ) & β’ πΆ = (IndMetβπ) & β’ π½ = (MetOpenβπΆ) & β’ πΎ = (topGenβran (,)) β β’ (π β NrmCVec β π β (π½ Cn πΎ)) | ||
| Theorem | nmcnc 29949 | The norm of a normed complex vector space is a continuous function to β. (For β, see nmcvcn 29948.) (Contributed by NM, 12-Aug-2007.) (New usage is discouraged.) |
| β’ π = (normCVβπ) & β’ πΆ = (IndMetβπ) & β’ π½ = (MetOpenβπΆ) & β’ πΎ = (TopOpenββfld) β β’ (π β NrmCVec β π β (π½ Cn πΎ)) | ||
| Theorem | smcnlem 29950* | Lemma for smcn 29951. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
| β’ πΆ = (IndMetβπ) & β’ π½ = (MetOpenβπΆ) & β’ π = ( Β·π OLD βπ) & β’ πΎ = (TopOpenββfld) & β’ π = (BaseSetβπ) & β’ π = (normCVβπ) & β’ π β NrmCVec & β’ π = (1 / (1 + ((((πβπ¦) + (absβπ₯)) + 1) / π))) β β’ π β ((πΎ Γt π½) Cn π½) | ||
| Theorem | smcn 29951 | Scalar multiplication is jointly continuous in both arguments. (Contributed by NM, 16-Jun-2009.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.) |
| β’ πΆ = (IndMetβπ) & β’ π½ = (MetOpenβπΆ) & β’ π = ( Β·π OLD βπ) & β’ πΎ = (TopOpenββfld) β β’ (π β NrmCVec β π β ((πΎ Γt π½) Cn π½)) | ||
| Theorem | vmcn 29952 | Vector subtraction is jointly continuous in both arguments. (Contributed by Mario Carneiro, 6-May-2014.) (New usage is discouraged.) |
| β’ πΆ = (IndMetβπ) & β’ π½ = (MetOpenβπΆ) & β’ π = ( βπ£ βπ) β β’ (π β NrmCVec β π β ((π½ Γt π½) Cn π½)) | ||
| Syntax | cdip 29953 | Extend class notation with the class inner product functions. |
| class Β·πOLD | ||
| Definition | df-dip 29954* | Define a function that maps a normed complex vector space to its inner product operation in case its norm satisfies the parallelogram identity (otherwise the operation is still defined, but not meaningful). Based on Exercise 4(a) of [ReedSimon] p. 63 and Theorem 6.44 of [Ponnusamy] p. 361. Vector addition is (1st βπ€), the scalar product is (2nd βπ€), and the norm is π. (Contributed by NM, 10-Apr-2007.) (New usage is discouraged.) |
| β’ Β·πOLD = (π’ β NrmCVec β¦ (π₯ β (BaseSetβπ’), π¦ β (BaseSetβπ’) β¦ (Ξ£π β (1...4)((iβπ) Β· (((normCVβπ’)β(π₯( +π£ βπ’)((iβπ)( Β·π OLD βπ’)π¦)))β2)) / 4))) | ||
| Theorem | dipfval 29955* | The inner product function on a normed complex vector space. The definition is meaningful for vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law. (Contributed by NM, 10-Apr-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) & β’ π = (Β·πOLDβπ) β β’ (π β NrmCVec β π = (π₯ β π, π¦ β π β¦ (Ξ£π β (1...4)((iβπ) Β· ((πβ(π₯πΊ((iβπ)ππ¦)))β2)) / 4))) | ||
| Theorem | ipval 29956* | Value of the inner product. The definition is meaningful for normed complex vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law, although for convenience we define it for any normed complex vector space. The vector (group) addition operation is πΊ, the scalar product is π, the norm is π, and the set of vectors is π. Equation 6.45 of [Ponnusamy] p. 361. (Contributed by NM, 31-Jan-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) & β’ π = (Β·πOLDβπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄ππ΅) = (Ξ£π β (1...4)((iβπ) Β· ((πβ(π΄πΊ((iβπ)ππ΅)))β2)) / 4)) | ||
| Theorem | ipval2lem2 29957 | Lemma for ipval3 29962. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) & β’ π = (Β·πOLDβπ) β β’ (((π β NrmCVec β§ π΄ β π β§ π΅ β π) β§ πΆ β β) β ((πβ(π΄πΊ(πΆππ΅)))β2) β β) | ||
| Theorem | ipval2lem3 29958 | Lemma for ipval3 29962. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) & β’ π = (Β·πOLDβπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((πβ(π΄πΊπ΅))β2) β β) | ||
| Theorem | ipval2lem4 29959 | Lemma for ipval3 29962. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) & β’ π = (Β·πOLDβπ) β β’ (((π β NrmCVec β§ π΄ β π β§ π΅ β π) β§ πΆ β β) β ((πβ(π΄πΊ(πΆππ΅)))β2) β β) | ||
| Theorem | ipval2 29960 | Expansion of the inner product value ipval 29956. (Contributed by NM, 31-Jan-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) & β’ π = (Β·πOLDβπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄ππ΅) = (((((πβ(π΄πΊπ΅))β2) β ((πβ(π΄πΊ(-1ππ΅)))β2)) + (i Β· (((πβ(π΄πΊ(iππ΅)))β2) β ((πβ(π΄πΊ(-iππ΅)))β2)))) / 4)) | ||
| Theorem | 4ipval2 29961 | Four times the inner product value ipval3 29962, useful for simplifying certain proofs. (Contributed by NM, 10-Apr-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) & β’ π = (Β·πOLDβπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (4 Β· (π΄ππ΅)) = ((((πβ(π΄πΊπ΅))β2) β ((πβ(π΄πΊ(-1ππ΅)))β2)) + (i Β· (((πβ(π΄πΊ(iππ΅)))β2) β ((πβ(π΄πΊ(-iππ΅)))β2))))) | ||
| Theorem | ipval3 29962 | Expansion of the inner product value ipval 29956. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) & β’ π = (Β·πOLDβπ) & β’ π = ( βπ£ βπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄ππ΅) = (((((πβ(π΄πΊπ΅))β2) β ((πβ(π΄ππ΅))β2)) + (i Β· (((πβ(π΄πΊ(iππ΅)))β2) β ((πβ(π΄π(iππ΅)))β2)))) / 4)) | ||
| Theorem | ipidsq 29963 | The inner product of a vector with itself is the square of the vector's norm. Equation I4 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (normCVβπ) & β’ π = (Β·πOLDβπ) β β’ ((π β NrmCVec β§ π΄ β π) β (π΄ππ΄) = ((πβπ΄)β2)) | ||
| Theorem | ipnm 29964 | Norm expressed in terms of inner product. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (normCVβπ) & β’ π = (Β·πOLDβπ) β β’ ((π β NrmCVec β§ π΄ β π) β (πβπ΄) = (ββ(π΄ππ΄))) | ||
| Theorem | dipcl 29965 | An inner product is a complex number. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (Β·πOLDβπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄ππ΅) β β) | ||
| Theorem | ipf 29966 | Mapping for the inner product operation. (Contributed by NM, 28-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (Β·πOLDβπ) β β’ (π β NrmCVec β π:(π Γ π)βΆβ) | ||
| Theorem | dipcj 29967 | The complex conjugate of an inner product reverses its arguments. Equation I1 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (Β·πOLDβπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (ββ(π΄ππ΅)) = (π΅ππ΄)) | ||
| Theorem | ipipcj 29968 | An inner product times its conjugate. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (Β·πOLDβπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((π΄ππ΅) Β· (π΅ππ΄)) = ((absβ(π΄ππ΅))β2)) | ||
| Theorem | diporthcom 29969 | Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (Β·πOLDβπ) β β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((π΄ππ΅) = 0 β (π΅ππ΄) = 0)) | ||
| Theorem | dip0r 29970 | Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (0vecβπ) & β’ π = (Β·πOLDβπ) β β’ ((π β NrmCVec β§ π΄ β π) β (π΄ππ) = 0) | ||
| Theorem | dip0l 29971 | Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (0vecβπ) & β’ π = (Β·πOLDβπ) β β’ ((π β NrmCVec β§ π΄ β π) β (πππ΄) = 0) | ||
| Theorem | ipz 29972 | 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.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (0vecβπ) & β’ π = (Β·πOLDβπ) β β’ ((π β NrmCVec β§ π΄ β π) β ((π΄ππ΄) = 0 β π΄ = π)) | ||
| Theorem | dipcn 29973 | Inner product is jointly continuous in both arguments. (Contributed by NM, 21-Aug-2007.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
| β’ π = (Β·πOLDβπ) & β’ πΆ = (IndMetβπ) & β’ π½ = (MetOpenβπΆ) & β’ πΎ = (TopOpenββfld) β β’ (π β NrmCVec β π β ((π½ Γt π½) Cn πΎ)) | ||
| Syntax | css 29974 | Extend class notation with the class of all subspaces of normed complex vector spaces. |
| class SubSp | ||
| Definition | df-ssp 29975* | Define the class of all subspaces of normed complex vector spaces. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.) |
| β’ SubSp = (π’ β NrmCVec β¦ {π€ β NrmCVec β£ (( +π£ βπ€) β ( +π£ βπ’) β§ ( Β·π OLD βπ€) β ( Β·π OLD βπ’) β§ (normCVβπ€) β (normCVβπ’))}) | ||
| Theorem | sspval 29976* | The set of all subspaces of a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| β’ πΊ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) & β’ π» = (SubSpβπ) β β’ (π β NrmCVec β π» = {π€ β NrmCVec β£ (( +π£ βπ€) β πΊ β§ ( Β·π OLD βπ€) β π β§ (normCVβπ€) β π)}) | ||
| Theorem | isssp 29977 | The predicate "is a subspace." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.) |
| β’ πΊ = ( +π£ βπ) & β’ πΉ = ( +π£ βπ) & β’ π = ( Β·π OLD βπ) & β’ π = ( Β·π OLD βπ) & β’ π = (normCVβπ) & β’ π = (normCVβπ) & β’ π» = (SubSpβπ) β β’ (π β NrmCVec β (π β π» β (π β NrmCVec β§ (πΉ β πΊ β§ π β π β§ π β π)))) | ||
| Theorem | sspid 29978 | A normed complex vector space is a subspace of itself. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.) |
| β’ π» = (SubSpβπ) β β’ (π β NrmCVec β π β π») | ||
| Theorem | sspnv 29979 | A subspace is a normed complex vector space. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.) |
| β’ π» = (SubSpβπ) β β’ ((π β NrmCVec β§ π β π») β π β NrmCVec) | ||
| Theorem | sspba 29980 | The base set of a subspace is included in the parent base set. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (BaseSetβπ) & β’ π» = (SubSpβπ) β β’ ((π β NrmCVec β§ π β π») β π β π) | ||
| Theorem | sspg 29981 | Vector addition on a subspace is a restriction of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ πΉ = ( +π£ βπ) & β’ π» = (SubSpβπ) β β’ ((π β NrmCVec β§ π β π») β πΉ = (πΊ βΎ (π Γ π))) | ||
| Theorem | sspgval 29982 | Vector addition on a subspace in terms of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ πΊ = ( +π£ βπ) & β’ πΉ = ( +π£ βπ) & β’ π» = (SubSpβπ) β β’ (((π β NrmCVec β§ π β π») β§ (π΄ β π β§ π΅ β π)) β (π΄πΉπ΅) = (π΄πΊπ΅)) | ||
| Theorem | ssps 29983 | Scalar multiplication on a subspace is a restriction of scalar multiplication on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = ( Β·π OLD βπ) & β’ π = ( Β·π OLD βπ) & β’ π» = (SubSpβπ) β β’ ((π β NrmCVec β§ π β π») β π = (π βΎ (β Γ π))) | ||
| Theorem | sspsval 29984 | Scalar multiplication on a subspace in terms of scalar multiplication on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = ( Β·π OLD βπ) & β’ π = ( Β·π OLD βπ) & β’ π» = (SubSpβπ) β β’ (((π β NrmCVec β§ π β π») β§ (π΄ β β β§ π΅ β π)) β (π΄π π΅) = (π΄ππ΅)) | ||
| Theorem | sspmlem 29985* | Lemma for sspm 29987 and others. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π» = (SubSpβπ) & β’ (((π β NrmCVec β§ π β π») β§ (π₯ β π β§ π¦ β π)) β (π₯πΉπ¦) = (π₯πΊπ¦)) & β’ (π β NrmCVec β πΉ:(π Γ π)βΆπ ) & β’ (π β NrmCVec β πΊ:((BaseSetβπ) Γ (BaseSetβπ))βΆπ) β β’ ((π β NrmCVec β§ π β π») β πΉ = (πΊ βΎ (π Γ π))) | ||
| Theorem | sspmval 29986 | Vector addition on a subspace in terms of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ πΏ = ( βπ£ βπ) & β’ π» = (SubSpβπ) β β’ (((π β NrmCVec β§ π β π») β§ (π΄ β π β§ π΅ β π)) β (π΄πΏπ΅) = (π΄ππ΅)) | ||
| Theorem | sspm 29987 | Vector subtraction on a subspace is a restriction of vector subtraction on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = ( βπ£ βπ) & β’ πΏ = ( βπ£ βπ) & β’ π» = (SubSpβπ) β β’ ((π β NrmCVec β§ π β π») β πΏ = (π βΎ (π Γ π))) | ||
| Theorem | sspz 29988 | The zero vector of a subspace is the same as the parent's. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
| β’ π = (0vecβπ) & β’ π = (0vecβπ) & β’ π» = (SubSpβπ) β β’ ((π β NrmCVec β§ π β π») β π = π) | ||
| Theorem | sspn 29989 | The norm on a subspace is a restriction of the norm on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (normCVβπ) & β’ π = (normCVβπ) & β’ π» = (SubSpβπ) β β’ ((π β NrmCVec β§ π β π») β π = (π βΎ π)) | ||
| Theorem | sspnval 29990 | The norm on a subspace in terms of the norm on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π = (normCVβπ) & β’ π = (normCVβπ) & β’ π» = (SubSpβπ) β β’ ((π β NrmCVec β§ π β π» β§ π΄ β π) β (πβπ΄) = (πβπ΄)) | ||
| Theorem | sspimsval 29991 | The induced metric on a subspace in terms of the induced metric on the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π· = (IndMetβπ) & β’ πΆ = (IndMetβπ) & β’ π» = (SubSpβπ) β β’ (((π β NrmCVec β§ π β π») β§ (π΄ β π β§ π΅ β π)) β (π΄πΆπ΅) = (π΄π·π΅)) | ||
| Theorem | sspims 29992 | The induced metric on a subspace is a restriction of the induced metric on the parent space. (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.) |
| β’ π = (BaseSetβπ) & β’ π· = (IndMetβπ) & β’ πΆ = (IndMetβπ) & β’ π» = (SubSpβπ) β β’ ((π β NrmCVec β§ π β π») β πΆ = (π· βΎ (π Γ π))) | ||
| Syntax | clno 29993 | Extend class notation with the class of linear operators on normed complex vector spaces. |
| class LnOp | ||
| Syntax | cnmoo 29994 | Extend class notation with the class of operator norms on normed complex vector spaces. |
| class normOpOLD | ||
| Syntax | cblo 29995 | Extend class notation with the class of bounded linear operators on normed complex vector spaces. |
| class BLnOp | ||
| Syntax | c0o 29996 | Extend class notation with the class of zero operators on normed complex vector spaces. |
| class 0op | ||
| Definition | df-lno 29997* | Define the class of linear operators between two normed complex vector spaces. In the literature, an operator may be a partial function, i.e., the domain of an operator is not necessarily the entire vector space. However, since the domain of a linear operator is a vector subspace, we define it with a complete function for convenience and will use subset relations to specify the partial function case. (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.) |
| β’ LnOp = (π’ β NrmCVec, π€ β NrmCVec β¦ {π‘ β ((BaseSetβπ€) βm (BaseSetβπ’)) β£ βπ₯ β β βπ¦ β (BaseSetβπ’)βπ§ β (BaseSetβπ’)(π‘β((π₯( Β·π OLD βπ’)π¦)( +π£ βπ’)π§)) = ((π₯( Β·π OLD βπ€)(π‘βπ¦))( +π£ βπ€)(π‘βπ§))}) | ||
| Definition | df-nmoo 29998* | Define the norm of an operator between two normed complex vector spaces. This definition produces an operator norm function for each pair of vector spaces β¨π’, π€β©. Based on definition of linear operator norm in [AkhiezerGlazman] p. 39, although we define it for all operators for convenience. It isn't necessarily meaningful for nonlinear operators, since it doesn't take into account operator values at vectors with norm greater than 1. See Equation 2 of [Kreyszig] p. 92 for a definition that does (although it ignores the value at the zero vector). However, operator norms are rarely if ever used for nonlinear operators. (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.) |
| β’ normOpOLD = (π’ β NrmCVec, π€ β NrmCVec β¦ (π‘ β ((BaseSetβπ€) βm (BaseSetβπ’)) β¦ sup({π₯ β£ βπ§ β (BaseSetβπ’)(((normCVβπ’)βπ§) β€ 1 β§ π₯ = ((normCVβπ€)β(π‘βπ§)))}, β*, < ))) | ||
| Definition | df-blo 29999* | Define the class of bounded linear operators between two normed complex vector spaces. (Contributed by NM, 6-Nov-2007.) (New usage is discouraged.) |
| β’ BLnOp = (π’ β NrmCVec, π€ β NrmCVec β¦ {π‘ β (π’ LnOp π€) β£ ((π’ normOpOLD π€)βπ‘) < +β}) | ||
| Definition | df-0o 30000* | Define the zero operator between two normed complex vector spaces. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.) |
| β’ 0op = (π’ β NrmCVec, π€ β NrmCVec β¦ ((BaseSetβπ’) Γ {(0vecβπ€)})) | ||
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