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User:KYN/WhyDualSpace

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Why do we need dual spaces?

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The concept of dual spaces is used frequently in abstact mathematics, but also has some practical applications. Consider a 2D vector space on which a differentiable function is defined. As an example, can be the Cartesian coordinates of points in a topographic map and can be the ground altitude which varies with the coordinate . According to theory, the infinitesimal change of at the point as a consequenece of changing the position an infintesimal amount is given by

the scalar product between the vector and the gradient of . Clearly, is a scalar and since it is constructed as a linear mapping on , by computing its scalar product with , it follows from the above defintion that is an element of .

From the outset, both vectors and can be seen as elements of . Why is a dual space needed? What is the difference between and in this case?

To see the difference between and , remember that in practice both vectors and must be expressed as a set of three real number which are their coordinates relative to some basis of . Intuitively we may choose to use an orthogonal basis, with normalized basis vectors which are mutually perpendicular. Let be a such a basis for . This means that can be written as

where are the (infinitesimal) coordinates of in the basis . Similiarly, can be written as

where are the coordinates of in the basis . Given that the coordinates of both