Functional Analysis Answers

Let X be the set of all real-valued functions x on the interval [0,1],
and let x≦y mean that x(t) ≦y(t) for all t∈[0,1]. Show that this
defines a partial ordering. Is it a total ordering? Does X have maximal
elements?

range of function cos(sinx)

If Z is an (n — l)dimensional subspace of an n-dimensioned vector
space X, show that Z is the null space of a suitable linear functional f
on X, which is uniquely determined to within a scalar multiple.

If Z is an (n — l)-dimension£il subspace of an n-dimensioned vector
space X, show that Z is the null space of a suitable linear functional f
on X, which is uniquely determined to within a scalar multiple.

(Linear extension) Let Z be a proper subspace of an n-dimensional
vector space X, and let f be a linear fimctional on Z Show that f can
be extended linearly to X, that is, there is a linear functioned f on X
such that f ̃|z =f

Funds flow analysis represents a stock to flow linkage.” – Justify

If $T\in\mathcal{B}(X,Y)$ is not compact can the restriction of $T$ to an infinite dimensional subspace of $X$ be compact?

If $T\in\mathcal{B}(X,Y)$ is not compact can the restriction of $T$ to an infinite dimensional subspace of $X$ be compact?

if T belongs to b(X,Y) is not compact can the restriction of T to an infinite dimensional subspace of X be compact?

What are the largest and smallest values of 2x + y on the circle x^2 + y^2 = 1? Where do these values occur? What does this have to do with eigenvectors and eigenvalues?