30-functionspace

30-functionspace - FUNCTION SPACES AND LINEAR MAPS Math...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: FUNCTION SPACES AND LINEAR MAPS Math 21b, O. Knill Homework: 4.2: 28,40,34,58,66,78* FUNCTION SPACES. Functions on the real line can be added f + g scaled f and contain a zero vector 0. P n , the space of all polynomials of degree n . The space P of all polynomials. C ( R ), the space of all smooth functions on the line C ( T ) the space of all 2 periodic functions. In all these function spaces, the function f ( x ) = 0 which is constantly 0 is the zero function. LINEAR TRANSFORMATIONS. A map T on a linear space X is called linear if T ( x + y ) = T ( x ) + T ( y ) , T ( x ) = T ( x ) and T (0) = 0. Examples are Df ( x ) = f ( x ) on C T f ( x ) = R x f ( x ) dx on C . T f ( x ) = f (2 x ). T f ( x ) = sin( x ) f ( x ) on C T f ( x ) = 5 f ( x ) T f ( x ) = f ( x- 1). SUBSPACES, EIGENVALUES, BASIS, KERNEL, IMAGE are defined as before X linear subspace f, g X, f + g X, f X, X . T linear transformation T ( f + g ) = T ( f ) + T ( g ) , T ( f ) = T ( f ) , T (0) = 0. f 1 , f 2 , ..., f n linear independent i c i f i = 0 implies f i = 0. f 1 , f 2 , ..., f n span X Every f is of the form i c i f i . f 1 , f 2 , ..., f n basis of X linear independent and span....
View Full Document

Ask a homework question - tutors are online