By Andrew H. Wallace

This self-contained therapy assumes just some wisdom of actual numbers and actual research. the 1st 3 chapters concentrate on the fundamentals of point-set topology, and then the textual content proceeds to homology teams and non-stop mapping, barycentric subdivision, and simplicial complexes. routines shape a vital part of the textual content. 1961 variation.

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Both embeddings L1 \ L1 Â L1 and L1 \ L1 Â L1 are continuous. 1) that L1 \ L1 is dense in L1 in the norm k kL1 , but it is not dense in L1 in the norm k kL1 . 2 The Space L01 The closure of L1 \L1 in L1 coincides with the space L01 , which can be described by means of decreasing rearrangements f as follows. 1 x>0 We set (Fig. 2 The Space L01 31 y y f * ( 0) f *( 0) f* f* f *( • ) > 0 f *( • ) = 0 f *( • ) 0 x 0 x Fig. L01 ; k kL01 / is a symmetric space. For this purpose, we need the following important property of decreasing rearrangements.

1) that L1 \ L1 is dense in L1 in the norm k kL1 , but it is not dense in L1 in the norm k kL1 . 2 The Space L01 The closure of L1 \L1 in L1 coincides with the space L01 , which can be described by means of decreasing rearrangements f as follows. 1 x>0 We set (Fig. 2 The Space L01 31 y y f * ( 0) f *( 0) f* f* f *( • ) > 0 f *( • ) = 0 f *( • ) 0 x 0 x Fig. L01 ; k kL01 / is a symmetric space. For this purpose, we need the following important property of decreasing rearrangements. 1. Let f and g be measurable functions.

X/ D ˆ 3x 2; x 2 23 ; 1 ; ˆ : x; x 2 Œ1; 1/; Transformations Â1 and Â2 are measure-preserving. m/. The functions f ; f1 ; f2 ; f3 are equimeasurable, f D f D f1 D f2 D f3 ; and fi ¤ fi ; i D 1; 2; 3. m/ such that f3 D f2 ı Â or f2 D f3 ı Â (see Fig. 6). 5. 1/. m/ such that kf g ı ÂkL1 \L1 < ": Proof. Let f and g satisfy the assumptions above. 1, we can construct equimeasurable step functions f" and g" such that kf f" kL1 \L1 Ä " and kg 2 y y y 1 1 1 0 1 x 0 " : 2 y 1 f2 f1 f g" kL1 \L1 Ä 1 x 0 f3 1/2 1 x 0 1/3 2/3 1 Fig.