A quick review of Mathe 114
... Calculus: Differential calculus + integral calculus 1. Establish functions for simple problems (Domain of a function - where the function is meaningful) 2. Properties of a function 1). Limit of a function when the variable approaches a finite point or +/- infinity 2). Continuity (continuous at a poi ...
... Calculus: Differential calculus + integral calculus 1. Establish functions for simple problems (Domain of a function - where the function is meaningful) 2. Properties of a function 1). Limit of a function when the variable approaches a finite point or +/- infinity 2). Continuity (continuous at a poi ...
1 Norms and Vector Spaces
... but there are of course vectors x ∈ ℓe for which the series doesn’t converge. Define ℓ2 to be those x for which it does converge ℓ2 = { x ∈ ℓe | kxk is finite } For example, the signal x(k) = ak is an element of ℓ2 if and only if |a| < 1. The perhaps surprising fact is that ℓ2 is a subspace of ℓe . ...
... but there are of course vectors x ∈ ℓe for which the series doesn’t converge. Define ℓ2 to be those x for which it does converge ℓ2 = { x ∈ ℓe | kxk is finite } For example, the signal x(k) = ak is an element of ℓ2 if and only if |a| < 1. The perhaps surprising fact is that ℓ2 is a subspace of ℓe . ...
Here
... exists a neighborhood U of a and V of f (a) such that 1) f maps U in a 1–1 manner onto V , 2) f −1 : V → U is differentiable at a, and 3) (f −1 )0 (f (a)) = 1/f 0 (a).” First note that being continuously differentiable is a key assumption. Otherwise the function f (x) = x + 2x2 sin(1/x) (for x 6= 0 ...
... exists a neighborhood U of a and V of f (a) such that 1) f maps U in a 1–1 manner onto V , 2) f −1 : V → U is differentiable at a, and 3) (f −1 )0 (f (a)) = 1/f 0 (a).” First note that being continuously differentiable is a key assumption. Otherwise the function f (x) = x + 2x2 sin(1/x) (for x 6= 0 ...
Calculus Jeopardy - Designated Deriver
... calculations of area using anti-derivatives. What is The Fundamental Theorem of Calculus? ...
... calculations of area using anti-derivatives. What is The Fundamental Theorem of Calculus? ...
2.7 Banach spaces
... spaces Rn , Cn , C[a, b], lp , and l∞ are the metric spaces introduced in Chapter 1. By Example 1.8.4 and Theorem 1.8.5, these metric spaces are complete. 2. This follows from Proposition 1.8.8 and the fact (Section 2.3) that the metric induced by the norm on M is the induced metric on X restricted ...
... spaces Rn , Cn , C[a, b], lp , and l∞ are the metric spaces introduced in Chapter 1. By Example 1.8.4 and Theorem 1.8.5, these metric spaces are complete. 2. This follows from Proposition 1.8.8 and the fact (Section 2.3) that the metric induced by the norm on M is the induced metric on X restricted ...
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... Definition Let (X, B, µ) be a measure space. Let 0 < p < ∞. The Lp -norm of a function f : X → C is defined as Z ||f ||p := ...
... Definition Let (X, B, µ) be a measure space. Let 0 < p < ∞. The Lp -norm of a function f : X → C is defined as Z ||f ||p := ...
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... The function space of rapidly decreasing functions S has the important property that the Fourier transform is an endomorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space of S, that is, for tempered distributions. Definition The ...
... The function space of rapidly decreasing functions S has the important property that the Fourier transform is an endomorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space of S, that is, for tempered distributions. Definition The ...