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12-functionspace

# 12-functionspace - FUNCTION SPACES/LINEAR MAPS Homework...

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FUNCTION SPACES/LINEAR MAPS, Math 21b, O. Knill Homework: Section 4.1 6-11,36,48,58,44*,12*-15* FROM VECTORS TO FUNCTIONS. Vectors can be displayed in different ways: 1 2 5 3 2 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 The values ( i,~v i ) can be interpreted as the graph of a function f : 1 , 2 , 3 , 4 , 5 , 6 R , where f ( i ) = ~v i . LINEAR SPACES. A space X in which we can add, scalar multiplications and where basic laws like commutativity, distributivity and associativity hold is called a linear space . Examples: Lines, planes and more generally, the n -dimensional Euclidean space. P n , the space of all polynomials of degree n . The space P of all polynomials. C , the space of all smooth functions on the line C 0 , the space of all continuous functions on the line. C 1 , the space of all differentiable functions on the line. C ( R 3 ) the space of all smooth functions in space.
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