l2 space

(See attached file for full problem description with symbols) --- Suppose is a matrix such that defines an element for . Show that . ---

Bounded linear functional

(See attached file for full problem description with proper symbols) --- Let be the set of all sequences , such that exists. Let be the dual space of , and consist of all functions , , such that for every is finite. Let , and define such that . Show that is a one-to-one, onto bounded linear functional on and that . Are and isometrically isomorphic? ---

Cauchy-Riemann equations, vector gradients and contour lines

1. a) The Cauchy-Riemann equation is the name given to the following pair of equations, ∂u/∂x=∂v/∂y and ∂u/∂y= -∂v/∂x which connects the partial derivatives of two functions u(x, y) and v(x, y) i) if u(x, y) =e^x cosy and v(x, y) =e^x siny, how do I prove that these functions satisfy the Cauchy-Riemann equations ii) if u(x, y) =½ In(x² +y²) and v(x, y) = tanֿ¹(y/x), how do I prove that theses functions satisfy the Cauchy-Riemann equations iii) if u and v are any functions that satisfy the Cauchy-Riemann equations, how do I prove that ∂²u/∂x² + ∂²u/∂y²=0 b... click for more

Functions And Notation

1. From the function f (x)=IxI How would I go about finding its image set using interval notation? 2. Again using interval notation, how would I go about finding the image set of the graph g(x)= Ix+3I -2 ? And how then would I go on to solve the equation g(x)=1, and discover if it had any geometrical significance? 3. How would I go about finding if there was solution to the equation g(x)=-4?

Domain and image sets

If I have the function f(x)=2e x+1 , how would I discover the domain and image set of function f? How would I go about finding the domain and image set of f -1? How would I go about solving the equation y= 2e x+1, to find x in terms of y?