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You have given a function λ : R → R with the following properties (x ∈ R, n ∈ N):

λ(n) = 0 , λ(x + 1) = λ(x) , λ(n+1/2)=1
Find two functions p, q : R → R with q(x)6=0 for all x such that λ(x) = q(x)(p(x) + 1).
Every Hilbert space has an orthonormal basis.
In a Banach space, show that every absolutely convergent series is convergent.
Let X is Banach space and Let M be a subspace of X. Than M is itself a Banach space (using the norm from X) if and only if M is closed.

Lex X be an inner product space over R. If x ,y X are such that ||x + y||=||x-y||then show that x  y.


Suppose that (xalpha)alpha element of j converges to x in X and (yalpha)alpha element of j converges to y in Y .show that (xalpha×yalpha) converges to x × y in X×Y
Consider the following examples in real world situation that describe a correspondence between phenomena:
-the income tax a person pays depends on the person's total income.
-the number of years of experience on a job has a direct bearing on the salary received.
-the amount by which your savings will grow in a year depends on the interest offered by the bank.
-the area of a circle depends on the radius of the circle.
-the distance travelled in a car at a certain rate depends on the time travelled.
A, express each situation in functional words. For example the income tax is a function of the total income.
B, for each situation, indicate the dependent and independent variable.
C, write the two variables as an ordered pair, and write the function using the notation y(x).
D, explain the meaning of a function in your own words, illustrate with your own specific example

prove that strong operator convergence implies weak operator convergence but converse is not true in general.



If a sub additive functional defined on a normed space X is nonnegative outside a sphere {x Illxll = r}, show that it is nonnegative for all x E X


prove that a finite partially ordered set A has at least one maximal element
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