Inner product space | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-27

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Inner_product_space


00:01:37 1 Definition
00:03:23 1.1 Alternative definitions, notations and remarks
00:06:18 1.2 Elementary properties
00:06:29 2 Examples
00:06:38 2.1 Real numbers
00:10:38 2.2 Euclidean space
00:10:57 2.3 Complex coordinate space
00:11:14 2.4 Hilbert space
00:13:36 2.5 Random variables
00:13:46 2.6 Real matrices
00:14:19 2.7 Vector spaces with forms
00:16:23 3 Norms on inner product spaces
00:17:49 4 Orthonormal sequences
00:19:51 5 Operators on inner product spaces
00:20:31 6 Generalizations
00:20:44 6.1 Degenerate inner products
00:21:06 6.2 Nondegenerate conjugate symmetric forms
00:21:53 7 Related products
00:22:25 8 See also
00:34:29 9 Notes
00:35:16 10 References
00:36:51 11 Sources
00:42:30 Operators on inner product spaces
00:43:22 ⟨x,y⟩ for all x, y in V, or equivalently, A is linear and ||Ax||
00:44:43 Generalizations
00:45:13 Degenerate inner products
00:45:51 0 does not imply x
00:46:12 V/{x : ||x||
00:46:51 Nondegenerate conjugate symmetric forms
00:48:46 Related products
00:51:28 See also



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SUMMARY
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In linear algebra, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors (zero inner product). Inner product spaces generalize Euclidean spaces (in which the inner product is the dot product, also known as the scalar product) to vector spaces of any (possibly infinite) dimension, and are studied in functional analysis. The first usage of the concept of a vector space with an inner product is due to Peano, in 1898.An inner product naturally induces an associated norm, thus an inner product space is also a normed vector space. A complete space with an inner product is called a Hilbert space. An (incomplete) space with an inner product is called a pre-Hilbert space, since its completion with respect to the norm induced by the inner product is a Hilbert space. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces.