Density / approximation argument

Suppose $f \in \mathcal{L}^{1}(\mathbb {R})$.

Suppose $U$ is a subspace of a normed vector space $V$. Suppose also that $W$ is a Banach space and $S: U \rightarrow W$ is a bounded linear map.

Suppose $\mu$ is a measure, $0 < p \leq \infty$, and $f \in \mathcal{L}^{p}(\mu )$. Prove that for every $\varepsilon > 0$, there exists a simple function $g \in \mathcal{L}^{p}(\mu )$ such that $\left\| f-g\right\|_{p} < \varepsilon$.

Suppose $0 < p < \infty$ and $f \in \mathcal{L}^{p}(\mathbb {R})$. Prove that for every $\varepsilon > 0$, there exists a step function $g \in \mathcal{L}^{p}(\mathbb {R})$ such that $\left\| f-g\right\|_{p} < \varepsilon$.

Suppose $0 < p < \infty$ and $f \in \mathcal{L}^{p}(\mathbb {R})$. Prove that for every $\varepsilon > 0$, there exists a continuous function $g: \mathbb {R} \rightarrow \mathbf{F}$ such that $\left\| f-g\right\|_{p} < \varepsilon$ and the set $\left\{ x \in \mathbb {R}: g(x) \neq 0\right\}$ is bounded.

Suppose $1 \leq p < \infty$ and $f \in \mathcal{L}^{p}(\mathbb {R})$.

Let $\varphi_{1}, \varphi_{2}, \ldots , \varphi_{n}, \ldots$ be nonnegative continuous functions on $[0,1]$ such that the limit

exists for every $k = 0,1,\ldots$. Show that the limit

exists for every continuous function $f$ on $[0,1]$.

Let $f$ and $g$ be continuous functions on $\mathbb {R}$ such that $f(x+1) = f(x)$, $g(x+1) = g(x)$, for all $x \in \mathbb {R}$. Prove that

Let $f$ be a continuous real valued function on $[0,1]$, and let the function $h$ in the complex plane be defined by

Let $F$ be a continuous complex valued function on the interval $[0,1]$. Let

for $z$ a complex number not in $[0,1]$.

Let $x \in L_{1}(\left[a, b\right])$. Prove that $\lim_{h \rightarrow 0+} \int_{a}^{b-h}\left|x(t+h)-x(t)\right| d t = 0$.

Let $f \in L_{\infty }(\mathbb {R})$, and there exists $a > 0$ such that $f = 0$ almost everywhere for $\left|x\right| \geq a$. Prove that there exists a sequence $\left\{ g_{n}\right\} \subset \mathcal{D}(\mathbb {R})$ such that $\left\| g_{n}\right\|_{\infty } \leq \left\| f\right\|_{\infty }, n \geq 1$, and $g_{n} \rightarrow f(\bmod m), n \rightarrow \infty$.

Let $X$ be an NLS, $M \subset X$ be a convex dense subset in $X$ and $f \in X^{*}$. Prove that the set $M \cap \operatorname {Ker} f$ is dense in $\operatorname {Ker} f$.

(Yamabe) Let $X$ be an NLS, $M \subset X$ be a convex dense subset in $X, x \in X, \varepsilon > 0, f_{1}, \ldots , f_{n} \in X^{*}$. Prove that $\exists y \in M,\left\| x-y\right\| < \varepsilon : f_{i}(y) = f_{i}(x), 1 \leq i \leq n$.

Let $X$ be an NLS, $Y$ be a Banach space, the linear set $M \subset X$ is dense in $X$, and $A \in \mathcal{L}(M, Y)$. Prove that the operator $A$ admits a unique extension $\bar{A} \in \mathcal{L}(X, Y)$, and $\left\| \bar{A}\right\| = \left\| A\right\|$.

In $L_{p}(\mathbb {R}), 1 \leq p < +\infty$, define the shift operator

Let $s_{n} \rightarrow s, n \rightarrow \infty$, and $s_{n} \neq s, n \geq 1$. Prove that $A_{s_{n}} \xrightarrow {s} A_{s}$, but $A_{s_{n}} \nRightarrow A_{s}$.

Let $H$ be a Hilbert space, $\left\{ A_{n}: n \geq 1\right\} \subset \mathcal{L}(H), \sup_{n \geq 1}\left\| A_{n}\right\| < +\infty$, and $\exists L \subset H, \bar{L} = H, \forall x, y \in L: \exists \lim_{n \rightarrow \infty }\left(A_{n} x, y\right)$.

Prove that there exists $A \in \mathcal{L}(H)$ such that $A_{n} \xrightarrow {w} A$.

Show that:

Prove that $\mathcal{D}$ is dense in each of the spaces

Prove that regular generalized functions are dense in $\mathcal{D}^{\prime }$.

Prove that regular generalized functions generated by compactly supported infinitely differentiable functions are dense in $\mathcal{D}^{\prime }$ (that is, $\mathcal{D}$ is dense in $\mathcal{D}^{\prime }$).

Prove that for any generalized function $F \in \mathcal{D}^{\prime }$ there exists a sequence of regular functions, generated by functions from $\mathcal{D}$, which converges to $F$ in $\mathcal{D}^{\prime }$. ¹

Let $\mathcal{D} = \left\{ f \in L^2(\mathbb {R}) : f \text{ is absolutely continuous on every bounded interval in } \mathbb {R} \text{ and } f' \in L^2(\mathbb {R})\right\}$. (Recall that for $f$ in $\mathcal{D}$, letting $Af = if'$ makes $A$ self-adjoint.)

Show that for every $f$ in $\mathcal{D}$ there is a sequence $\left\{ f_n\right\}$ in $C_c^{(1)}(\mathbb {R})$ such that $f_n \to f$ and $f_n' \to f'$ in $L^2(\mathbb {R})$.

Recall Proposition 6.13: if $\psi \in L^1(\mathbb {R})$ is such that $(2\pi )^{-1/2}\int_{\mathbb {R}} \psi (x) \, dx = 1$ and if for $\varepsilon > 0$, $\psi_\varepsilon (x) = \varepsilon^{-1}\psi (x/\varepsilon )$, then for every $f$ in $C_0(\mathbb {R})$, $\psi_\varepsilon * f(x) \to f(x)$ uniformly on $\mathbb {R}$.

Let $\psi$ and $\psi_\varepsilon$ be as above. If $f \in L^p(\mathbb {R})$, $1 \leq p < \infty$, show that $\left\| f * \psi_\varepsilon - f\right\|_p \to 0$ as $\varepsilon \to 0$. If $f \in L^{\infty }(\mathbb {R})$, show that $f * \psi_\varepsilon \to f$ (weak*).

Let $\left\{ x_{n}\right\}$ be a sequence in a Hilbert space $H$, and let $E$ be a dense subset of $H$. Show that if there exist some $A, B > 0$ such that $A\left\| x\right\|^{2} \leq \sum \left| \left\langle x, x_{n} \right\rangle \right|^{2} \leq B\left\| x\right\|^{2}$ for $x \in E$, then $\left\{ x_{n}\right\}$ is a frame for $H$. Thus it suffices to establish the frame condition on some dense (and hopefully "nice") subset of $H$ (compare hypothesis (b) of Theorem 7.4).

Show that $\left\{ \psi \in L^{2}(\mathbb {R}): \psi \text{ is admissible}\right\}$ is dense in $L^{2}(\mathbb {R})$.

The existence of infinitely differentiable, compactly supported functions is demonstrated in Exercise 11.9. Combine Exercises 13.13 and 13.15 with Theorem 13.13 to show that $C^{\infty }(\mathbf{T})$ is dense in $L^{p}(\mathbf{T})$ for $1 \leq p < \infty$, and is dense in $C(\mathbf{T})$ with respect to the uniform norm. Conclude that $C^{m}(\mathbf{T})$ is dense in these spaces for each integer $m \geq 0$.

Prove that $\mathcal D(\Omega )$ is dense in $\mathcal E(\Omega )$ for any domain $\Omega \subset \mathbb {R}^{n}$.

Find the orthogonal complements in $L^{2}\left({(1,0)}\right)$ of the following sets:

Let $f \in C(\left[0,1\right])$. Prove that

Let $f \in C(\left[0,1\right])$ and $\int_{0}^{1} f(x) g^{\prime \prime }(x) d x=0$ for every function $g \in C^{2}(\left[0,1\right])$ vanishing in neighborhoods of the points 0 and 1. Prove that $f$ is a linear function.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $f$ be a nonnegative function that is integrable over $X$, and let $\varepsilon >0$. Prove that there exists a set $A \in \mathfrak {M}$ such that $\mu (A)<\infty$ and a simple function $s$ that vanishes outside of $A$ and satisfies $0 \leq s \leq f$ and $\int_{X}|f-s| d \mu <\varepsilon$.

Let $X, Y$ be Banach spaces, let $T \in \mathcal{L}(X, Y)$, let $\left\{ T_{n}\right\}$ be a bounded sequence in $\mathcal{L}(X, Y)$, and suppose that $\lim_{n}\left\| T_{n} x-T x\right\| =0$ for all $x$ in a dense subset of $X$. Prove or disprove: $\lim_{n}\left\| T_{n}-T\right\| =0$.

Let $f(x)$ denote a properly integrable function on $\left[a, b\right]$. There exist to every positive number $\varepsilon$ two step-functions, $\psi (x)$ and $\Psi (x)$, such that for the entire interval $\left[a, b\right]$

and

It is even possible to choose $\psi (x)$ and $\Psi (x)$ so that their points of discontinuity are equidistant.

The limit relation in Problem 155 is also valid if $f(x)$ denotes a properly integrable function on $\left[0, c\right]$.

Show that the subspace of simple functions is dense in $L^{\infty }(E)$.

Let $\mathcal{S}$ be a dense subset of $X$. Show that if $\Phi \in X^{*}$, then $\left\| \Phi \right\|_{*} = \sup \left\{ \Phi (f) \mid f \in \mathcal{S},\left\| f\right\| \leq 1\right\}$.

With the notations of Exercise 11, suppose $f \in \mathscr {R}(\alpha )$ and $\varepsilon > 0$. Prove that there exists a continuous function $g$ on $[a, b]$ such that $\left\| f-g\right\|_{2} < \varepsilon$.

Hint: Let $P = \left\{ x_{0}, \ldots , x_{n}\right\}$ be a suitable partition of $[a, b]$, define

if $x_{i-1} \leq t \leq x_{i}$.

Assume $f \in \mathscr {R}(\alpha )$ on $[a, b]$, and prove that there are polynomials $P_{n}$ such that

(Compare with Exercise 12, Chap. 6).

A step function is, by definition, a finite linear combination of characteristic functions of bounded intervals in $\mathbb {R}^{1}$. Assume $f \in L^{1}\left(\mathbb {R}^{1}\right)$, and prove that there is a sequence $\left\{ g_{n}\right\}$ of step functions so that

Suppose $\left\{ \Lambda_{n}\right\}$ is a sequence of bounded linear transformations from a normed linear space $X$ to a Banach space $Y$, suppose $\left\| \Lambda_{n}\right\| \leq M < \infty$ for all $n$, and suppose there is a dense set $E \subset X$ such that $\left\{ \Lambda_{n} x\right\}$ converges for each $x \in E$. Prove that $\left\{ \Lambda_{n} x\right\}$ converges for each $x \in X$.

Let $f: \mathbb {R} \rightarrow \overline{\mathbb {R}}$ be such that $\int_{\mathbb {R}} f \, d\bar{\lambda }$ exists, is finite. Further suppose that $\int_{(a,b)} f \, d\bar{\lambda }=0$ for every $-\infty <a<b<+\infty$. Then $f=0$ $\bar{\lambda }$-a.e. on $\mathbb {R}$. [Work from indicators to simple functions, then to nonnegative functions, then to general integrable functions.]

Here we will show that the collection $C_{c}$ of continuous functions with compact support is dense in $L^{1}$, where the notion of distance is as given in the previous exercise. That is, given $f \in L^{1}$ and $\varepsilon >0$, there is $g \in C_{c}$ such that $\left\| f-g\right\|_{1}<\varepsilon$. Work in the following stages.

The following is an outline of an alternative proof that $\mathfrak {C}_{c}$ is a dense subset of $L^{p}\left(\mathbb {R}^{k}, \mathscr {B}^{k}, \lambda_{k}\right)$ (where $1 \leqslant p<+\infty$) that doesn't use Lusin's Theorem [as was used in Claim 3]. Here $\mathfrak {C}_{c}$ denotes the collection of real-valued functions on $\mathbb {R}^{k}$ that are continuous and have compact support, and $\mathfrak {S}_{0}$ denotes the collection of $\mathcal{F}/\mathfrak {B}^{*}$-measurable finite-valued simple functions $s$ on $\Omega$ such that $\mu \left(\left\{ \omega \in \Omega : s(\omega ) \neq 0\right\} \right)<\infty$.

Claim 3: Given any bounded $f \in L^{p}\left(\mathbb {R}^{k}, \mathscr {B}^{k}, \lambda_{k}\right)$ and $\varepsilon >0$, we can find $g \in \mathfrak {C}_{c}$ such that $\left\| f-g\right\|_{p}<\varepsilon$ and $\sup_{\omega \in \mathbb {R}^{k}}\left|g(\omega )\right| \leqslant \sup_{\omega \in \mathbb {R}^{k}}\left|f(\omega )\right|$. In particular, $\mathfrak {C}_{c}$ is a dense subset of $L^{p}\left(\mathbb {R}^{k}, \mathscr {B}^{k}, \lambda_{k}\right)$.

Claim 2: $\mathfrak {S}_{0}$ is a dense subset of $L^{p}$, where $\mathfrak {S}_{0}$ denotes the collection of $\mathcal{F}/\mathfrak {B}^{*}$-measurable finite-valued simple functions $s$ on $\Omega$ such that $\mu \left(\left\{ \omega \in \Omega : s(\omega ) \neq 0\right\} \right)<\infty$.

We have shown for a general $(\Omega , \mathcal{F}, \mu )$ that $\mathfrak {S}_{0}$ is a dense subset of $L^{p}$ for $1 \leqslant p<+\infty$ (Claim 2 above).

Show that $B$ in Example 3.3.1 (c) is a basis for $H$.

Let $g: \mathbb {R} \rightarrow \mathbb {R}$ be infinitely differentiable and be zero outside a bounded interval.