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	Reference.com/Encyclopedia/Inner product space
		Inner products allow the rigorous introduction of intuitive geometrical notions such as the angle between vectors or length of vectors in spaces of all dimensions.
		Inner product spaces generalize Euclidean spaces (with the dot product as the inner product) and are studied in functional analysis.
		Using the norm associated to the inner product, one has the notion of dense subset, and the appropriate definition of orthonormal basis is that the algebraic span (subspace of finite linear combinations of basis vectors) should be dense.
	www.reference.com /browse/wiki/Inner_product_space (1907 words)

	Inner product space - Wikipedia, the free encyclopedia
		In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors.
		Inner product spaces generalize Euclidean spaces (with the dot product as the inner product) and are studied in functional analysis.
		Hence, the inner product is a Hermitian form.
	en.wikipedia.org /wiki/Inner_product_space (1808 words)

	Product (mathematics) - Wikipedia, the free encyclopedia
		In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied.
		When matrices or members of various other associative algebras are multiplied the product usually depends on the order of the factors; in other words, matrix multiplication, and the multiplications in those other algebras, are non-commutative.
		The dot product and cross product are forms of multiplication of vectors; the same as dot product or more general are the scalar product and the inner product; see also Inner product space
	en.wikipedia.org /wiki/Product_(mathematics) (182 words)

	Dot product - Wikipedia, the free encyclopedia
		In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity.
		As the cosine of 90° is zero, the dot product of two perpendicular vectors is always zero.
		The inner product generalizes the dot product to abstract vector spaces, it is normally denoted by .
	en.wikipedia.org /wiki/Dot_product (1018 words)

	PlanetMath: inner product
		The standard example of an inner product is the dot product on
		Every inner product space is a normed vector space, with the norm being defined by
		This is version 11 of inner product, born on 2002-01-24, modified 2006-10-22.
	www.planetmath.org /encyclopedia/InnerProduct.html (143 words)

	PlanetMath: inner product space
		With this norm, an inner product space is also a normed vector space.
		If this metric is complete then the inner product space is called a Hilbert space.
		This is version 12 of inner product space, born on 2002-01-24, modified 2005-11-25.
	planetmath.org /encyclopedia/InnerProductSpace.html (119 words)

	Complete Inner Product Space
		The inner product of a function with itself is the norm.
		The Schwarz inequality in quantum mechanics is analogous to dot products and cosines in Euclidean space.
		In a similar fashion, it is hoped that because the product of a transpose of a quaternion with a quaternion has the properties of a complete inner product space, the power of the mathematical field of quaternions can be used to solve a wide range of problems in quantum mechanics.
	world.std.com /~sweetser/quaternions/quantum/bracket/bracket.html (774 words)

	PlanetMath: inner product
		The standard example of an inner product is the dot product on
		Every inner product space is a normed vector space, with the norm being defined by
		This is version 10 of inner product, born on 2002-01-24, modified 2002-07-25.
	planetmath.org /encyclopedia/InnerProduct.html (142 words)

	Comparing vectors - the inner product
		In n-dimensions the inner product is defined by the natural extension of (7)
		We therefore now examine the significance of the inner product in 2 dimensions.
		Essentially the inner product tells us how well `aligned' two vectors are.
	www.shef.ac.uk /psychology/gurney/notes/l2/subsection3_2_2.html (262 words)

	Inner Product Fractals from Fuzzy Logics
		This fractal may be constructed using J inner products [3].
		Fuzzy logics may be used on this m in the same construction as the inner product fractals discussed earlier with a slight variation to accommodate the fuzzy “or” and “and” operators.
		As before, this is substituted into the inner product formula and the rest is the same as the probabilistic logic.
	www.vector.org.uk /archive/v192/coxe192.htm (1162 words)

	Amazon.com: Inner Product Spaces and Applications: Books: T M Rassias (Site not responding. Last check: 2007-10-08)
		Contributing authors deal primarily with the interaction among problems of analysis and geometry in the context of inner product spaces by presenting new and old characterizations of inner product spaces among normal linear spaces and the use of such spaces in various research problems of pure and applied mathematics.
		Contributing authors deal primarily with the interaction among problems of analysis & geometry in the context of inner product spaces by presenting new & old characterizations of inner product spaces among normal linear spaces & the use of such spaces in various research problems of pure & applied mathematics.
		In a real inner product space the orthogonality of two vectors can be expressed in different ways by means of the norm induced by the scalar product.
	www.amazon.com /exec/obidos/tg/detail/-/0582317118?v=glance (619 words)

	Linear Algebra (Math 2318) - Vector Spaces - Inner Product Spaces (Site not responding. Last check: 2007-10-08)
		Example 1 The Euclidean inner product as defined in the Euclidean n-space section is an inner product.
		This formula is very similar to the Euclidean inner product formula and so showing that this is an inner product will be almost identical to showing that the Euclidean inner product is an inner product. There are differences, but for the most part it is pretty much the same.
		We’ll leave it to you to verify that this is an inner product. It should be fairly simple if you’ve had calculus and you followed the verification of the weighted Euclidean inner product. The key is again the fact that the weight is a strictly positive function on the interval
	tutorial.math.lamar.edu /AllBrowsers/2318/InnerProductSpaces.asp (803 words)

	An Investigation of the Characteristics of the inner product calculation in high speed signal analysis -- from ...
		In the present investigation, the characteristics of the integrated inner product calculation as a function of Doppler frequency present in signal analysis have been explored.
		The integrated inner product is an estimate of the distance measurement in space or time between the reference signal and the target signal.
		The integrated inner product distance measurement as a function of scaling, translation and frequency is shown graphically.
	library.wolfram.com /infocenter/Articles/831 (158 words)

	The Inner Product - Home
		So with The Inner Product I set out to write about technical subjects at the edges of professional game developers' understanding (or at least my own personal understanding), and to perform experiments that may be useful to professional programmers but also out-of-the-ordinary enough that people would not have explored those directions themselves.
		The "inner product" is a mathematical operation, also commonly known as the "dot product"; this operation takes two vectors, which can be quite large many-dimensional quantities, and it crunches them together and yields for you a simple scalar value, telling you something about the relationship between those vectors.
		Secondly, the name "inner product" signifies that we're building the mechanisms deep inside a game; whereas the players of a game see mostly the outer parts of the product (texture maps, 3D meshes and the like), it's that inner engine that makes it all possible.
	number-none.com /product (608 words)

	Inner Product Fractals
		Using J notation, the ordinary matrix product is denoted by +/.
		Several of those involved a product, base 2, of a matrix of indices with a generalized matrix product.
		The "matrix product" is generalized in the sense that the "sum" and "pairwise product" operations in ordinary matirx arithmetic are replaced by other functions.
	ww2.lafayette.edu /~reiterc/mvp/ipf (285 words)

	Level Curves (Site not responding. Last check: 2007-10-08)
		The inner product is an operation that operates on two vectors to produce a scalar result.
		One is that the dot product of a vector with itself produces the square of that vector's magnitude.
		Integral Inner Product: If F and G are vectors representing the functions f(x) and g(x), the integral inner product F.G is the definite integral of f(x)g(x) over the interval b1..b2 with respect to x.
	www.math.unl.edu /Dept/Resources/Projects/Multimedia/curvefitting.html (1606 words)

	Linear Algebra (Part 1.1) ~ 3DSoftware.com
		The Scalar Product is produced by processing two vectors that have the same number of components (cells).
		In linear algebra, the terms scalar product, inner product and dot product mean the same thing, and are used interchangeably.
		The term inner product is particularly descriptive in our discussion here, since the multiplication is among internal parts (components) of the vectors, and – as we shall see later – works inside matrices.
	www.3dsoftware.com /Math/Programming/LinAlg01 (387 words)

	Inner Product Space (Site not responding. Last check: 2007-10-08)
		In simple terms, the inner product measures the relative alignment between two vectors.
		The inner products above are the "usual" choices for those spaces.
		The inner product naturally defines a norm: (∥x∥,≔,\) though not every norm can be defined from an inner product.
	cnx.org /content/m10430/2.12 (288 words)

	Mathematical Structure -- Inner Product Spaces (Site not responding. Last check: 2007-10-08)
		Because we want to be able to use the dot product to carry geometric ideas and tools over to situations where the geometry is not immediately evident, we want to study the dot product algebraically and then use it motivated by the connection between the algebra and the geometry in R
		Prove that C[a, b] with this operation is an inner product space -- that is, prove that this operation satisfies the dot product properties.
		Prove that if u, v, and w are vectors in an inner product space and u is perpendicular to w and v is perpendicular to w and s and t are two real numbers then s u + t v is also perpendicular to w.
	math.montana.edu /frankw/ccp/multiworld/building/dotproduct/refer.htm (1194 words)

	Perpendicularity in Vector Spaces (Site not responding. Last check: 2007-10-08)
		To discuss orthonormal bases for other vector spaces, we must extend the idea of dot product to what is called an inner product.
		Vector spaces with an inner product are also called inner product spaces.
		Several properties of inner products are generalizations of properties of the dot product in
	distance-ed.math.tamu.edu /Math640/chapter3/node13.html (470 words)

	Inner product (Site not responding. Last check: 2007-10-08)
		Also, I will denote the inner product by rather than with circular brackets as it done in the question.
		Determine whether g is an inner product on R^3.
		I mean the inner product isn't necessarily the dot product.
	www.physicsforums.com /showthread.php?p=781875#post781875 (1215 words)

	inner product - FOLDOC Definition (Site not responding. Last check: 2007-10-08)
		Attention is seldom paid to any other kind of inner product.
		An inner product, g: V -> V', is said to be positive definite iff, for all non-zero v in V, (gv)v > 0; likewise negative definite iff all such (gv)v = 0; negative semi-definite or non-positive definite iff all such (gv)v <= 0.
		Where only one inner product enters into discussion, it is generally elided in favour of some piece of syntactic sugar, like a big dot between the two vectors, and practitioners don't take much effort to distinguish between vectors and their duals.
	www.nightflight.com /foldoc-bin/foldoc.cgi?inner+product (170 words)

	Inner product (Site not responding. Last check: 2007-10-08)
		The scalar product (or dot product or inner product) of two row vectors (or two inner product...
		The standard inner product on R^n is precisely the dot product of.
		Grassmann derives the concept of inner product from that of.Noting that is linear in both E and F, Grassmann calls the inner product of.
	www.cafesystems.com /inner+product.html (257 words)

	The Inner Product
		The inner product (or ``dot product'') is an operation on two vectors which produces a scalar.
		Adding an inner product to a Banach space produces a Hilbert space (or ``inner product space'').
		The complex conjugation of the second vector is done in order that a norm will be induced by the inner product:
	www.technick.net /public/code/cp_dpage.php?aiocp_dp=guide_dft_inner_product (218 words)

	Dot Product
		Sometimes an appropriate inner product, or dot product is defined.
		When scalars are taken from the field of real numbers, the "standard" dot product of two vectors is the sum of the products of the corresponding entries.
		Their dot product, divided by n, approximates the integral of fg.
	www.mathreference.com /la,dot.html (656 words)

	Inner Circle: Product Catalog
		The following roleplaying products are available for the popular d20 System produced by Wizards of the Coast.
		Here you will be able to find all information about a product in one central location.
		Violet Dawn and The Inner Circle are trademarks of The Inner Circle.
	icirclegames.com /product_catalog.html (2048 words)

	Inner Product (Site not responding. Last check: 2007-10-08)
		What is a natural inner product to define on the space of all functions f:S->R? I want to approximate an arbitrary function with a polynomial of a fixed degree (both of which are defined only on S), and I want to use projections to do it, but I have no inner product.
		I want to use the fact that the orthogonal projection of a vector in the space of polynomials of degree less than k to the space of finitely-supported real-valued functions is the best approximation.
		The expression he wants to minimize looks like it's the dot product of something with itself, but that doesn't seem to be a useful observation, although it might help a little with bookkeeping.
	www.physicsforums.com /showthread.php?t=114148 (1009 words)

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