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No. Let $(X, \cal A, \mu)$ be $[0,1]$ with Lebesgue measure. Let $E = L^2[0,1]$ with inner product $\langle \alpha,\beta\rangle := \int \alpha(t)\overline{\beta(t)}\;dt$. Define $f : [0,1] \to L^2[0,1 …

For example: For the real Banach space $L^p(\mathbb R)$, with $1 < p < \infty$, the "conjugate space" is $L^q(\mathbb R)$ where $\frac{1}{p}+\frac{1}{q}=1$. For general linear functional $T$ on $L^p$ …

An example. $X = [0,1]$ with the usual metric. $\mathcal C[0,1] = \mathcal C_b[0,1] = \mathcal C_0[0,1]$. $\mathcal C[0,1]^* = \mathcal M[0,1]$. Let $\mu_n$ be the unit point-mass at $1/n$ and $\mu …

comment I think they are not isometric, having different structure for the set of extreme points. The set of extreme points for the unit ball of $\|\cdot\|_\infty$ is a torus: $$T = \{(z_1,z_2) : |z_1 …

Let's try this, as a negative answer. It has been a long time since I seriously worked on non-standard analysis—so criticism is welcome. I follow mostly the terminology of Robinson's book Non-Standa …

Take $\mathcal X = L^1(\Omega,\mu)$ where $\Omega = \{1,2,3,\dots\}$ and measure $\mu(\{k\}) = p_k$ with $p_k > 0$, $\sum p_k = 1$. The norm in $\mathcal X$ is $$ \|f\|_{\mathcal X} = \sum_k |f(k)|p_ …

Example 3 of almost-periodic functions. It is more natural than the description given in the OP. Harald Bohr defined almost periodic functions in order to study Dirichlet series as applicable in anal …

A counterexample for random variable with nonseparable range. Let $\omega_1$ be the smallest uncountable ordinal. Let $\Omega = [0,\omega_1)$, the set of countable ordinals, with the order topology. …

With the definitions in the OP, this is false. It is OK if the Banach space $B$ is separable and $(\Omega,\mathcal F, P)$ is an arbitrary probability space. It is OK if the Banach space $B$ is arbit …

Let's begin with an example, which supports the conjecture: not inner product, map from a group, satisfies the norm inequalities, but not a homomorphism. $V = l^1$, the Banach space of sequences $\ …

Certainly "linearly independent" is not good enough. Example. Let $e_n$ be an orthonormal basis in a Hilbert space. Take $$ x_n = \frac{1}{\sqrt{n}}(e_1+\dots +e_n) $$ Coefficients are chosen so …

Not an answer to this problem. For a non-reflexive $G$ see example: Exercise (23.32) in Hewitt & Ross, Abstract Harmonic Analysis I (Springer 1963). Consider a topological vector space $L^p(\mathbb …

Let's try this. I think this is false, since the total variation has no connection with the distance $d$. (That other question does not seem to mention the total variation norm at all.) counterexam …

The answer is sometimes "no", even when $Y$ is one-dimensional. Let's say the scalars are $\mathbb R$; the same argument will work for $\mathbb C$. Let $\omega_1$ be the least uncountable ordina …

In an $L^1(\mu)$ space, we can tell when two elements are disjoint (that is, have disjoint support): $f_1, f_2$ are disjoint if and only if $\|f_1\pm f_2\| = \|f_1\|+\|f_2\|$. I claim that if $L_1(\m …