Для того чтобы

In order that $f^*$ be (но не is) a good approximation to a given function $f$, we require the error function $f-f^*$ to be small in some sense

For a function $f$ to be continuous it is necessary that ...

A necessary and sufficient condition for a matrix to be nonsingular is that its determinant be nonzero

In order that this process have (но не has) meaning, it is necessary that it give (но не gives) a unique result

Formula (1) is applied to study the above case (to derive the theorem below, to obtain an $x$ with norm not exceeding 1)

Let us consider some examples to show how this function decreases at infinity

This approach is too complicated to be used in the above case

This particular case is important enough to be considered separately

We now apply (use) Theorem 1 to obtain $x=y$

Insert (1) into (2) (substitute (1) into (2)) to find that ...

We partially order $Z$ by declaring $X<Y$ to mean that ...

For this to happen (in order that this happens), this set must be compact

For the second estimate to hold, it is enough to assume that ...

Then for such a map to exist, we should assume that ...

One must use basis functions of degree at least two in order for $x$ to be nonzero