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In Mathematics , an inner product space is a Vector Space with additional structure, an '''inner product''' (also called '''scalar product'''), which allows us to introduce geometrical notions such as Angle s and Length s of vectors. Inner product spaces generalize Euclidean Space s (with the Dot Product as the inner product) and are studied in Functional Analysis . An inner product space is sometimes also called a ''pre-Hilbert space'', since its Completion with respect to the Metric Induced by its inner product is a Hilbert Space . Inner product spaces were referred to as unitary spaces in earlier work, although this terminology is now rarely used. DEFINITIONS In the following article, the Field of Scalar s denoted F is either the field of Real Number s R or the field of Complex Number s '''C'''. See below. Formally, an inner product space is a vector space ''V'' over the field F together with a Positive-definite Nondegenerate Sesquilinear Form , called an ''inner product''. For real vector spaces, this is actually a positive-definite nondegenerate Symmetric Bilinear Form . Thus the inner product is a map :$\backslash langle\; \backslash cdot,\; \backslash cdot\; angle\; :\; V\; imes\; V\; ightarrow\; \backslash mathbf\{F\}$ satisfying the following Axiom s: Conjugate symmetry: :: :This condition implies that $\backslash langle\; x,x\; angle\; \backslash in\; \backslash mathbf\{R\}$ for all $x\; \backslash in\; V$, because $\backslash langle\; x,x\; angle\; =\; \backslash overline\{\backslash langle\; x,x\; angle\}$. } , as is the Conjugate Transpose .)'' Sesquilinear ity: :: :: :By combining these with conjugate symmetry, we get: :: :: Nonnegativity: :: (This makes sense because $\backslash langle\; x,x\; angle\; \backslash in\; \backslash mathbf\{R\}$ for all $x\backslash in\; V$.) Nondegeneracy: given by $x\backslash mapsto\; \backslash langle\; x,\backslash cdot\; angle$ is an isomorphism. For a Finite-dimensional vector space, it suffices to check Injectivity : :: :Hence, the inner product is a Hermitian Form . The property of an inner product space $V$ that ::$\backslash langle\; x+y,z\; angle=\; \backslash langle\; x,z\; angle+\; \backslash langle\; y,z\; angle$ and $\backslash langle\; x,y+z\; angle\; =\; \backslash langle\; x,y\; angle\; +\; \backslash langle\; x,z\; angle$ for all $x,\; y,\; z\; \backslash in\; V$ is known as ''additivity''. Note that if F='''R''', then the conjugate symmetry property is simply ''symmetry'' of the inner product, i.e. :: $\backslash langle\; x,y\; angle=\backslash langle\; y,x\; angle.$ In this case, sesquilinearity becomes standard Linear ity.
	:<math> \x\	\sqrt{\langle x, x angle}</math>
	::<math> 0 \leq \langle X -\lambda Y, X -\lambda Y Angle	\langle x, x angle - \langle y , y angle^{-1} \langle x,y angle^2 </math>

  ::<math> \r \cdot X\ r \cdot \ x\</math>
  :Because Of The Triangle Inequality And Because Of Axiom 2, We See That · Is A Norm Which Turns ''V'' Into A "http://wwwinformationdelightinfo/encyclopedia/entry/normed_vector_space" class="copylinks">Normed Vector Space and hence also into a Metric Space The most important inner product spaces are the ones which are Complete with respect to this metric they are called Hilbert Space s Every inner product ''V'' space is a Dense subspace of some Hilbert space This Hilbert space is essentially uniquely determined by ''V'' and is constructed by completing ''V''
  ::<math> \x + Y\^2 + \x - Y\^2 2\x\^2 + 2\y\^2 </math>
  ::<math> \x\^2 + \y\^2 \x+y\^2 </math>
  ::<math> \sum {i 1}^n \x_i\^2 = \left\\sum_{i=1}^n x_i ight\^2 </math>
  ::<math> \sum {i 1}^\infty\x_i\^2 = \left\\sum_{i=1}^\infty x_i ight\^2, </math>


is an isometric linear map ''V'' → ''l''² with a dense image.

This theorem can be regarded as an abstract form of Fourier Series , in which an arbitrary orthonormal basis plays the role of the sequence of Trigonometric Polynomial s. Note that the underlying index set can be taken to be any countable set (and in fact any set whatsoever, provided ''l²'' is defined appropriately, as is explained in the article Hilbert Space ).
In particular, we obtain the following result in the theory of Fourier series:

Theorem. Let ''V'' be the inner product space $C${Link without Title} . Then the sequence (indexed on set of all integers) of continuous functions
:$e\_k(t)\; =\; (2\; \backslash pi)^\{-1/2\}e^\{i\; k\; t\}$
is an orthonormal basis of the space $C${Link without Title} with the ''L''² inner product. The mapping
:
is an isometric linear map with dense image.

Orthogonality of the sequence {e_k}_k follows immediately from the fact that if k ≠ j, then
:$\backslash int\_\{-\backslash pi\}^\backslash pi\; e^\{-i\; (j-k)\; t\}\; \backslash ,\; dt\; =\; 0$
Normality of the sequence is by design, that is, the coefficients are so chosen so that the norm comes out to 1. Finally the fact that the sequence has a dense algebraic span, in the ''inner product norm'', follows from the fact that the sequence has a dense algebraic span, this time in the space of continuous periodic functions on $${Link without Title} with the uniform norm. This is the content of the Weierstrass theorem on the uniform density of trigonometric polynomials.


OPERATORS ON INNER PRODUCT SPACES

Several types of Linear maps ''A'' from an inner product space ''V'' to an inner product space ''W'' are of relevance:
  Isometries, Ie ''A'' Is Linear And <''Ax'', ''Ay''> <''x'', ''y''> for all ''x'', ''y'' in ''V'', or equivalently, ''A'' is linear and ''Ax'' = ''x'' for all ''x'' in ''V'' All isometries are Injective Isometries are Morphism s between inner product spaces, and morphisms of real inner product spaces are orthogonal transformations (compare with Orthogonal Matrix )
  Then The Function ''x'' <''x'',&nbsp''x''>^1/2 makes sense and satisfies all the properties of norm except that ''x'' = 0 does not imply ''x'' = 0 (Such a functional is then called a Semi-norm ) We can produce an inner product space by considering the
  Quotient ''W'' ''V''/{&nbsp''x''&nbsp:&nbsp''x''&nbsp=&nbsp0} The sesquilinear form <&nbsp,&nbsp> factors through ''W''