Spaces of Riemann-integrable functions

Prove that if a sequence of functions $\left(x_{n}(t)\right)$ converges uniformly to a function $x(t)$ on the segment $[a ; b]$ and $x_{n}-x \in R L_{p}[a ; b]$, then this sequence also converges to the function $x$ in the seminorm of the space $R L_{p}[a ; b]$ (see example 2).

Construct an example of a sequence of continuous nonnegative functions on the segment $[0 ; 1]$ that converges in the sense of the norm of the space $C L_{p}[0 ; 1]$ (see problem 66) but does not converge at any point of this segment.

A set $E$ is called dense in a seminormed space $X$ if for every point $x \in X$ and every $\varepsilon > 0$ there exists a point $y \in E$ such that $\left\| y-x\right\| < \varepsilon$. The set $C_{0}^{\infty }(a ; b)$, $-\infty \leqslant a < b \leqslant +\infty$ (see problem 113 of $\S 18$), is dense in the space $R L_{p}(a ; b), 1 \leqslant p < +\infty$ (see example 2).

If $1 \leqslant p < q < +\infty$, then the set $R L_{q}[a ; b]$ is a dense subspace of the space $R L_{p}[a ; b]$ (see example 2).

Prove the stated assertion. The following inclusions hold:

Prove the stated assertion. The system of powers $1, t, t^{2}, \ldots , t^{n}, \ldots$ is complete in the space $C L_{p}[a ; b]$ (see problem 66).

Prove the stated assertion.

Prove the stated assertion. In the linear space $R L_{2}(a ; b),-\infty \leqslant a < b \leqslant +\infty$, consisting of real functions whose square is integrable (generally, in the improper sense) on the interval $(a ; b)$, the functional

is an almost-scalar product and is not a scalar product.

Prove the stated assertion. In the linear space of functions $x:[a ; b] \rightarrow R$ continuous on the segment $[a ; b]$, the functional

is a scalar product. The resulting space with the scalar product is denoted $C L_{2}[a ; b]$.

Prove the stated assertion. In the linear space consisting of complex-valued functions whose squared modulus is integrable on a finite or infinite interval, the functional

is an almost-scalar product and is not a scalar product.

Prove the stated assertion. In the space $C L_{2}(a ; b)$ of real functions continuous on a finite or infinite interval $(a ; b),-\infty \leqslant a < b \leqslant +\infty$, whose square is integrable on this interval, the functional

is a scalar product.

Find the angles of the triangle with vertices at the points $x_{1}(t) \equiv 0$, $x_{2}(t) \equiv 1, x_{3}(t) \equiv t$ in the space $C L_{2}[-1 ; 1]$ (see problem 11 of § 20).

If in the space $R L_{2}(a ; b)$ (see problem 10) the series $\sum_{n = 1}^{\infty } x_{n}(t)$ converges and its sum equals $x(t)$, i.e., $\sum_{n = 1}^{\infty } x_{n}(t) = x(t)$, then for every function $\varphi (t) \in R L_{2}[a ; b]$ the equality

holds; in particular, for a finite interval $(a, b)$ the equality

holds, i.e., the given series can be integrated term by term.

The space $C L_{2}[a ; b]$ (see problem 11) is not a Hilbert space.

If $\lim_{n \rightarrow \infty } \int_{a}^{b}\left[x(t)-x_{n}(t)\right]^{2} d t = 0$, then the sequence of functions $\left\{ x_{n}(t)\right\}$ is called convergent in the mean-square sense on the segment $[a ; b]$ to the function $x(t)$.

The function $x(t) = 1 / \sqrt{t}$ is not the limit, in the mean-square sense on the segment $[0 ; 1]$, of a sequence of continuous functions.