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Unformatted text preview: 12 Lecture 3 Vector Spaces of Functions 3.1 Space of Continuous Functions C ( I ) Let I ⊆ R be an interval. Then I is of the form (for some a < b ) I = { x ∈ R  a < x < b } , an open interval; { x ∈ R  a ≤ x ≤ b } , a closed interval; { x ∈ R  a ≤ x < b } { x ∈ R  a < x ≤ b } . Recall that the space of all functions f : I→ R is a vector space. We will now list some important subspaces: Example (1) . Let C ( I ) be the space of all continuous functions on I . If f and g are continuous, so are the functions f + g and rf ( r ∈ R ). Hence C ( I ) is a vector space. Recall, that a function is continuous, if the graph has no gaps. This can be formulated in different ways: a) Let x ∈ I and let ² > 0 . Then there exists a δ > 0 such that for all x ∈ I ∩ ( x δ,x + δ ) we have  f ( x ) f ( x )  < ² This tells us that the value of f at nearby points is arbitrarily close to the value of f at x . 13 14 LECTURE 3. VECTOR SPACES OF FUNCTIONS b) A reformulation of (a) is: lim x → x f ( x ) = f ( x ) 3.2 Space of continuously differentiable func tions C 1 ( I ) Example (2) . The space C 1 ( I ). Here we assume that I is open. Recall that f is differentiable at x if lim x → x f ( x ) f ( x ) x x = lim h → f ( x + h ) f ( x ) h =: f ( x ) exists. If f ( x ) exists for all x ∈ I , then we say that f is differentiable on I . In this case we get a new function on I x 7→ f ( x ) We say that f is continuously differentiable on I if f exists and is con tinuous on I ....
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This note was uploaded on 11/23/2011 for the course MATH 2025 taught by Professor Staff during the Spring '08 term at LSU.
 Spring '08
 Staff
 Vector Space

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