We have to solve for x.

Solve for x: 4*+6* = 9* P.S * equals the X.

Verifying an Inner Product for Continuous Functions

Please see the attached file for the fully formatted problems. Suppose f(x) and g(x) are continuous real-valued functions defined for [0,1]. Define vectors in n, F= ( f(x1), f(x2), …,f(xn)) and G= g(x1), g(x2), …,g(xn)), where xk = k/n. Why is n = 1/n  f(xk) g(xk) dx not an inner product for the space of continuous functions?

Topology : Open Unit Balls

Please see the attached file for the fully formatted problems. Let , and denote the three metrics defined on . What are the open unit balls , and with respect to these three metrics? Make a sketch and describe them algebraically.

Counter Images of a Function

Please see the attached file for the fully formatted problems.

Topologies : Open Sets

• Let X:={a,b,c} be a set of three elements. A certain topology of X contains (among others) the sets {a}, {b}, and {c}. List all open sets in the topology T. • Let X’:={a,b,c,d,e} be a set of five elements. A certain topology T’ on X’ contains (among others) the sets {a,b,c}, {c,d} and {e}. List any other open set in T’ which you know must be there. Also name one subset of X’ which may or may not be open. --- Please see the attached file for full problem description.