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4 votes
Accepted

Let $E$ be a Banach space, $p \in (1, \infty)$, and $L_p := L_{p}(X, \mu, E)$. Is $(L_p)^* \cong L_{p'}$ where $\frac{1}{p} + \frac{1}{p'} = 1$?

4 votes
Accepted

Textbook on TVS that contains Theorem 32.2 about the completeness of the space of continuous linear maps

3 votes
Accepted

Brezis' exercise 6.25(i): the existence of $M,P \in \mathcal L(E)$ such that $M \circ(I+K)=I-P$

3 votes
Accepted

If $x_n \rightharpoonup x$ and $|x_n| \to |x|$, then $x_n \to x$

2 votes
Accepted

Can Brezis's Ex 3.9 be generalized to arbitrary subset $M$ of $E$?

2 votes
Accepted

Let $E$ be a reflexive Banach space and $M$ its closed linear subspace. Then $M$ is reflexive

2 votes
Accepted

The Borel $\sigma$-algebra of a metric space is the smallest $\sigma$-algebra with respect to which all continuous functionals are measurable

2 votes
Accepted

The space of bounded Lipschitz continuous functions is dense in that of bounded uniformly continuous functions

2 votes
Accepted

The space of finite signed measures with the total variation norm is a Banach space

2 votes
Accepted

Brezis' exercise 8.24.2: do we need the assumption that $k$ is sufficiently large?

2 votes
Accepted

How to show the continuity of the Laplacian of the heat kernel (in t)

2 votes
Accepted

Knott-Smith optimality

2 votes
Accepted

The closed unit ball of a normed space $X$ is compact if and only if $X$ is finite-dimensional

2 votes
Accepted

Characterize subdifferential of a convex function by directional derivative

2 votes
Accepted

Disintegration theorem: how to construct the $\nu$-null set $E$ and pick the collection $\{\tilde{\pi}_{\sharp} f \mid f \in C(X)\}$?

1 vote
Accepted

Disintegration theorem: how is $\mu_y$ a probability measure for $\nu$-a.e. $y\in Y$?

1 vote
Accepted

Strong convergence of subgradients of a convex Fréchet differentiable function on a normed space

1 vote
Accepted

The map $\varphi : E \to \mathbb R \cup \{+\infty\}$ is proper convex l.s.c. if and only if $\varphi = \varphi^{**}$

1 vote
Accepted

Assume $A$ is open convex and $f$ convex continuous. Then $f$ is $L$-lipschitz on $A$ if and only if $\partial f(A) \subset L B_{X^*}$

1 vote
Accepted

Disintegration theorem: how to obtain $\|\pi_\sharp f\|_{L^\infty(Y)}\leq \|f\|_{C(X)}$ for all $f\in C(X)$?

1 vote
Accepted

Disintegration theorem: how do the authors prove that $\mu_y$ is supported on $\pi^{-1} (y)$ for $\nu$-a.e. $y \in Y$?

1 vote

How to prove that $\mu_n \rightharpoonup \mu$ IFF $\mu_n \overset{*}{\rightharpoonup} \mu$ and $\mu_n (X) \to \mu (X)$?

1 vote

Let $X$ be locally compact separable and $\mu_n\overset{*}{\rightharpoonup}\mu$. Then $|\mu|(O)\le\liminf_n|\mu_n|(O)$ for all open subsets $O$ of $X$

1 vote

A theorem about whether or not the topology on $X$ is the initial topology induced by some family of maps on $X$

1 vote
Accepted

Assume $A$ is open convex and $f$ convex continuous. Then $f$ is Gâteaux differentiable at $a \in A$ if and only if $\partial f (a)$ is a singleton

1 vote
Accepted

Are the "bounded, uniformly continuous functions" analogous to test functions as seen in Schwartz Distributions?

1 vote
Accepted

How the requirement of Itô's lemma is satisfied in this theorem about integration by parts?

1 vote

Prove that if $t:=\lim_n t_n >0$, then $(u_n)$ converges

1 vote
Accepted

If $E \subset F$ then $\dim(V/F) \le \dim (V/E)$

1 vote
Accepted

Is there $(f_n)\subset C^\infty_c (I)$ such that $\|f_n - 1_{[a, b]} \|_{L^2 (I)} \to 0$ and $\| f_n'\|_{L^2 (I)} \to 0$?

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