Real Analysis

(i) The upper Riemann sum of a real bounded function defined on [a, b] is:

(ii) If a curve γ in R^k is rectifiable, then :

(iii) Suppose Lim_n→∞ f_n(x) = f(x) ∀x ∈ E Put M_k = sup_n≥k |f_n(x) - f(x)|

(iv) Then f_n → f uniformly on set E if and only if:

(v) Let X be a vector space. Then a Linear transformation of X into X is called:

(v) Radius of convergence R of the power series ∑ C_n Zⁿ,

where C_n = nⁿ is:

Let f be a bounded function and α a monotonically increasing function on [a, b]. Then prove that:

If f is continuous on [a, b] then prove that f ∈ R(α) on [a, b].

Define Integration of vector-valued functions.

Write a short note on rectifiable curves also define its length.

Prove that the sequence of functions {f_n} defined on E, converges uniformly on E, if and only if for every ε > 0 there exists an integer N such that m ≥ N, n ≥ N, x ∈ E implies |f_m(x) - f_n(x)| ≤ ε.

Show that the sequence {f_n}, where f_n(x) = x / (1+nx²)

Let A ∈ L(Rⁿ, Rⁿ). Then prove that ||A|| < ∞ and A is a uniformly continuous mapping of Rⁿ into Rⁿ.

Let E be an open set in Rⁿ, f maps E into Rⁿ and x ∈ E and Lim_h→0 (f(x+h) - f(x) - A_h) / |h| = 0 holds with A = A_x, and with A = A_x. Then prove that A_x = A_x.

Suppose f is defined in an open set E ⊂ R², and D₁f and D₂f exist at every point of E. Suppose Q ⊂ E is a closed rectangle with sides parallel to the coordinate axes, having (a, b) and (a + h, b + k) as opposite vertices (h ≠ 0, k ≠ 0). Put Δf(Q) = f(a + h, b + k) - f(a + h, b) - f(a, b + k) + f(a, b). Then prove that there is a point (x, y) in the interior of Q such that Δf(Q) = hk (D₁D₂ f) (x, y).

Determine the radius of convergence of the power series.

Let f be a bounded function and α be a monotonically increasing function on [a, b]. Then prove that f ∈ R(α) on [a, b] if and only if for every ε > 0 there exists a partition p such that U(p, f, α) - L(p, f, α) < ε.

Let f ∈ R on [a, b]. For a ≤ x ≤ b put F(x) = ∫_a^x f(t) dt. Then prove that F is continuous on [a, b]; furthermore, if f is continuous at a point x₀ of [a, b], prove that F is differentiable at x₀ and F'(x₀) = f(x₀).

If γ is continuous on [a, b] then prove that γ is rectifiable and ∧(γ) = ∫_a^b |γ'(t)| dt.

Let ∑ a_n be a series of real numbers which converges, but not absolutely. Suppose −∞ < α ≤ β < ∞. Then prove that there exists a rearrangement ∑ a'_n with partial sums s'_n such that lim inf s'_n = α, lim sup s'_n = β.

Suppose f_n uniformly on a set E in a metric space. Let x be a Limit point of E, and Suppose that lim_t→x f_n(t) = A_n (n = 1, 2, 3, ...) Then prove that {A_n} converges and lim_n→∞ A_n = lim_t→x f_n(t).

Let α be a monotonically increasing on [a, b]. Suppose f_n ∈ R(α) on [a, b], for n = 1, 2, 3, ......, and suppose f_n → f uniformly on [a, b]. Then prove that f ∈ R(α) on [a, b] and ∫_a^b f dα = lim_n→∞ ∫_a^b f_n dα.

A Linear operator A on a finite-dimensional vector space X is one-to-one if and only if the range of A is all of X.

Suppose E is an open set in Rⁿ, f maps E into R^m. f is differentiable at x₀ ∈ E, g maps an open set containing f(E) into R^k and g is differentiable at f(x₀). Then prove that the mapping F of E into R^k defined by F(x) = g(f(x)) is differentiable at x₀ and F'(x₀) = g'(f(x₀)) f'(x₀).

Let ∑ a_n xⁿ be a power series with unit radius of convergence and let f(x) = ∑_n=0^∞ a_n xⁿ. If the series ∑ a_n converges then prove that lim_x→1^- f(x) = ∑_n=0^∞ a_n.

Let f be a C¹-mapping of an open set E ⊂ Rⁿ into Rⁿ, such that f(a) = b for some point (a, b) ∈ E. If we put A = f'(a), B = f'(b) and assume that A₁ is invertible, then there exist open sets U ⊂ Rⁿ and W ⊂ Rⁿ with a ∈ U and b ∈ W, having the following property:

To every y ∈ W there corresponds a unique x such that x ∈ U and f(x) = y.