By C. Davis, B. Grünbaum, F.A. Sherk

Geometry has been outlined as that a part of arithmetic which makes entice the feel of sight; yet this definition is thrown unsure through the lifestyles of serious geometers who have been blind or approximately so, similar to Leonhard Euler. occasionally it sounds as if geometric equipment in research, so-called, consist in having recourse to notions outdoors these it seems that correct, in order that geometry has to be the becoming a member of of not like strands; yet then what let's imagine of the significance of axiomatic programmes in geometry, the place connection with notions outdoor a limited reper tory is banned? no matter what its definition, geometry sincerely has been greater than the sum of its effects, greater than the results of a few few axiom units. it's been an important present in arithmetic, with a particular strategy and a distinc ti v e spirit. A present, in addition, which has now not been consistent. within the Thirties, after a interval of pervasive prominence, it seemed to be in decline, even passe. those similar years have been these during which H. S. M. Coxeter was once starting his medical paintings. Undeterred by means of the unfashionability of geometry, Coxeter pursued it with devotion and suggestion. via the Fifties he seemed to the wider mathematical global as a consummate practitioner of a weird, out-of-the-way artwork. this day there isn't any longer whatever that out-of-the-way approximately it. Coxeter has contributed to, exemplified, shall we virtually say presided over an unanticipated and dra matic revival of geometry.

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**Example text**

C sgn (7 0 u) . (7 o uES(r) T . AT(z). 28) Therefore S,(T'(V)) c P'(V), A,(T'(V)) c A'(V). Furthermore, it is easy t o show that a symmetric tensor is invariant under the symmetrizing mapping and an alternating tensor is invariant under the alternating mapping. Therefore P T ( V )= S,(P'(V)), A'(V) = A,(A'(V)). Thus P'(V) = S,(TT(V)), A'(V) = A,(T'(V)). 0 The above discussion about symmetric and alternating contravariant tensors can be applied analogously t o covariant tensors. The set of all symmetric covariant tensors of order r is denoted by P'(V*),and the set of all alternating covariant tensors of order r by A r ( V * ) .

If V is an n-dimensional vector space over IF, then V' is also an ndimensional vector space over IF. To see this, suppose {ul , .. ,a,} is a basis of V , and n v = CV%Ei v, f E V'. i= 1 Then i=l Therefore the linear function f is determined by its values f ( u i ) , 1 5 i 5 n, on the basis. We may define linear functions a" E V ' , 1 5 i 5 n , such that Then a*i(w) = vi. 4) says that any element in V * can be expressed as a linear combination of {a*i, 1 5 i 5 n}. It is easy to see that the expression is unique, and therefore {a*', 1 5 i 5 n } is a basis of V * ,which is called the dual basis of {ai, 1 5 i 5 n}.

Therefore V can be viewed as a vector space formed by all IF-valued linear functions on V * . In other words, V is the dual space of V * . Now we generalize the discussion above. Assume that V ,W,2 are all finitedimensional vector spaces over the field IF. l. 1 + a2w2) =alf(w1) + a2f(w2). J , a h 1 a 2 2 . 2 ) = a l f ( v ,W l ) a 2 f ( v w2). , + Similarly we can define an r-linear map f : V1 x . . x V, are vector spaces over IF. 9) + 2 where V1,. . 1 become IF-valued linear functions, IF-valued bilinear functions, and IF-valued r-linear functions, respectively.