mmtheorems84 - Metamath Proof Explorer

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**Statement List for Metamath Proof Explorer -** 8301-8400 - Page 84 of 108
Type	Label	Description
Statement

Theorem	nvcni3 8301	Epsilon-delta property of a continuous operator. (Normed complex vector space version of metcni 7864.)
Base IndMet IndMet Open Open NrmCVec $/\$ NrmCVec $/\$ Cn $/\$ $/\$ $/\$ $/\$

Theorem	nvlmcl 8302	Closure of the limit of a converging vector sequence.
Base IndMet NrmCVec $/\$

Theorem	nvlmle 8303	If the norm of each member of a converging sequence is less than or equal to a given amount, so is the norm of the convergence value.
Base IndMet NrmCVec $/\$ $/\$ $/\$ $/\$

Theorem	cnims 8304	The metric induced on the complex numbers. cnmet 7874 proves that it is a metric. (Contributed by Steve Rodriguez, 5-Dec-2006; revised by nm 15-Jan-2008.)
IndMet

Theorem	sqcn 8305	The square function on complex numbers is continuous.
IndMet Open $/\$ Cn

Theorem	sqcn2 8306	The square function on complex numbers is continuous.
Open $/\$ Cn

Theorem	nmcnilem 8307	Lemma for nmcni 8308.

Theorem	nmcni 8308	The norm of a normed complex vector space is a continuous function.
IndMet Open Open NrmCVec Cn

Theorem	nmcn 8309	The norm of a normed complex vector space is a continuous function.
IndMet Open Open NrmCVec Cn

Theorem	nmcn2 8310	The norm of a normed complex vector space is a continuous function to .
IndMet Open NrmCVec Cn topGen (,)

Theorem	nmcn3 8311	The norm of a normed complex vector space is a continuous function to .
IndMet Open OpenIndMet NrmCVec Cn

Theorem	nmcnc 8312	The norm of a normed complex vector space is a continuous function to . (For , see nmcn 8309.)
IndMet Open Open NrmCVec Cn

Theorem	abscn 8313	The absolute value function on complex numbers is continuous.
Open Open Cn

Theorem	abscncfALT 8314	Absolute value is continuous. Alternate proof of abscncf 7246.


Theorem	va1cnlem 8315	Lemma for va1cn 8316.

Theorem	va1cn 8316	Vector addition is continuous in its first operand.
Base IndMet Open $/\$ NrmCVec Cn

Theorem	sm1cnilem 8317	Lemma for sm1cni 8318.

Theorem	sm1cni 8318	Scalar multiplication is continuous in its first operand.
Base IndMet Open Open $/\$ NrmCVec Cn

Inner product

Syntax	cip 8319	Extend class notation with the class inner product functions.


Definition	df-ip 8320	Define a function that maps a complex normed 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 , the scalar product is , and the norm is .
NrmCVec $/\$ $/\$ $/\$

Theorem	ipval2lem1 8321	Lemma for ipval3 8329.

Theorem	ipfval 8322	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.
Base NrmCVec $/\$ $/\$

Theorem	ipval 8323	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 , the scalar product is , the norm is , and the set of vectors is . Equation 6.45 of [Ponnusamy] p. 361.
Base NrmCVec $/\$ $/\$

Theorem	ipval2lem2 8324	Lemma for ipval3 8329.

Theorem	ipval2lem3 8325	Lemma for ipval3 8329.

Theorem	ipval2lem4 8326	Lemma for ipval3 8329.

Theorem	ipval2 8327	Expansion of the inner product value ipval 8323. Warning: The HTML proof page is 0.5MB in size.
Base NrmCVec $/\$ $/\$

Theorem	4ipval2 8328	Four times the inner product value ipval3 8329, useful for simplifying certain proofs.
Base NrmCVec $/\$ $/\$

Theorem	ipval3 8329	Expansion of the inner product value ipval 8323.
Base NrmCVec $/\$ $/\$