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If for some realthen substituting shows that the inequality is equivalent to and clearly equality holds if a is non.

So since the integrand is always non-negative it must be that its discriminant is negative. If there is a basis of and numbers such thatprove that is angle preserving if and only if all are equal.

## Calculo En Variedades/ Calculus of Variations

Equality holds precisely when one is a nonnegative multiple of the other. In particular, has an inverse. This is a consequence of the analogous assertion of the next problem. So T is Since is a linear map of a finite dimensional vector space into itself, it follows that is also onto. Let be cqlculo preserving.

The case where is treated similarly. Let be an orthogonal basis of. Further, giventhere is a withand so. Then impliesi. Show that is norm preserving if and only if is spivsk product preserving. If and in are both non-zero, then the angle between anddenotedis defined to be which makes sense by Theorem 2. No, you could, for example, vary at discrete points without changing the values of the integrals.

Define to be the linear transformation such that is angle preserving, the are also pairwise orthogonal. On the other hand, ifthen the result follows The inequality follows from Theorem If and are continuous, then the assertion is true.

This completes the characterization. Similarly, if is norm preserving, then the polarization identity together with the linearity of T give:. Prove that if is norm preserving, then variedadee angle preserving.

Note, however, that the equality condition does not follow from a. The angle preserving are precisely those which ccalculo be expressed in the form where U is angle preserving of the kind in part bV is norm preserving, and the operation is functional composition.

The linear transformation is angle preserving if is and for1. Geometrically, if, and are the vertices spivai a triangle, then the inequality says that the length of a side is no larger than the sum of the lengths of the other two sides.

I claim that equality holds precisely when one vector is a non-positive multiple of the other. Further, is norm preserving since.

Parte 1 de For the converse, suppose that is angle preserving. A linear transformation is called norm preserving if1. Clearly, any of this form is angle variedade as the composition of two angle preserving linear transformations is angle preserving.

### Calculo En Variedades/ Calculus of Variations : Michael Spivak :

Then part a gives the inequality of Theorem 2. Further, maps each to a scalar multiple of itself; so is a map of the variedadee in part b. In fact, suppose that for eachthere is an with.

Trabalho do Professor Shing Tung Yau. So, this condition suffices to make be angle preserving.

If for some realthen substituting back into the equality shows that must be non-positive or must variedadws 0. To correct the situation, add the condition that the be pairwise orthogonal, because all the cross terms are zero.

The transformation is by Cramer’s Rule because the determinant of its matrix is 1. Then the inequality holds true in an open neighborhood of since and are.

The first assertion is the triangle inequality.