DR.VIJAYABALAN

  Hilbert spaces [ edit ] Main article:  Hilbert space The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red). Complete inner product spaces are known as  Hilbert spaces , in honor of  David Hilbert . [51]  The Hilbert space  � 2 ( Ω ) ,  with inner product given by ⟨ �   ,   � ⟩ = ∫ Ω � ( � ) � ( � ) ¯ � � , where  � ( � ) ¯  denotes the  complex conjugate  of  � ( � ) , [52] [nb 11]  is a key case. By definition, in a Hilbert space any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions  � �  with desirable properties that approximates a given limit function, is equally crucial. Early analysis, in the guise of the  Taylor approximation , established an approximation of  differentiable functions   �  by polynomials. [53]  By the  Stone–Weierstrass theorem , ever...

Linear algebra 90 languages Article Talk Read Edit View history From Wikipedia, the free encyclopedia In three-dimensional  Euclidean space , these three planes represent solutions to linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations. Linear algebra  is the branch of  mathematics  concerning  linear equations  such as: � 1 � 1 + ⋯ + � � � � = � , linear maps  such as: ( � 1 , … , � � ) ↦ � 1 � 1 + ⋯ + � � � � , and their representations in  vector spaces  and through  matrices . [1] [2] [3] Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of  geometry , including for defining basic objects such as  lines ,  planes  and  rotations . Also,  functional analysis , a branch of mathematical analysis, may be viewe...