# Questions tagged [functional-analysis]

Functional analysis, the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

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### Let $X$ be a Banach Space.Prove that, $X$ is strictly convex iff every points of $S(X)$ is exposed points of $B(X)$.

Some definitions- $X$ is said to be strictly convex or rotund if for all $x,y\in S(X),\ x\ne y$ we have $\left\lVert\frac{x+y}{2}\right\rVert<1$ A point $x_0\in B(X)$ is said to be a exposed point ...
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### How to prove that the convergence of $(f_n)$ is uniform on compact sets?

I'm reading Theorem 0.9 in this lecture note. Below is my attempt where I got stuck at the end. Could you elaborate on how to finish the proof? Let $C$ be an open convex subset of a Banach space $X$....
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### Showing that a sequence of compact operators converge uniformly to their pointwise limit

I am working out of Phillipe G. Ciarlet's Linear and Nonlinear Functional Analysis with Applications and am struggling with exercise 4.9-5. The problem is as follows: Let $G$ be a function in the ...
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### $X$ is strictly convex (rotund) iff for all $x,y\in S(X)$ with $x\ne y$ we have $\lVert x+y\rVert <2$

A Banach Space $X$ is said to be strictly convex or rotund if for all $x,y\in X$ we have $\lVert x+y\rVert<\lVert x\rVert+\lVert y\rVert$ unless $x,y$ are multiple of each other. We have to prove ...
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### Weak Convergence for a continuous function without compact support

Let $X$ be an open subset of $\mathbb R^n$ and let $\Omega$ be a relatively compact, open subset of $X$. Let $\{ \mu_n\}$ be a sequence of positive measures that converge weakly to the positive ...
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### Uniformly convexity of a space

Let function $\delta(\varepsilon)=\inf\{1 -\frac{\|x+y\|}{2}:x,y\in B_{1},\|x-y\|\geq \varepsilon \}$ be a modulus of convexity of Banach space $X$. How to prove that the space is uniformly convex ...
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### Does my proof generalize this exercise about continuity of convex lower semi-continuous function?

I'm trying to solve below question (Proposition 0.7.) in this lecture note. Let $C$ be an open convex subset of a normed space $X$ and $f: C \to \mathbb{R}$ convex. (a) If $f$ is u.s.c., then $f$ is ...
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### Almost sure convergence for lipschitz functions

Let $x_n \to x$($x_n$ sequence of random variables) s.t $\sum \mathbb{E}|f(x_n) - f(x)| < \infty$. For any $f$ Lipschitz and bounded. Then $x_n \to x$ almost sure. My attempt: As series converge, ...
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### Apply Arzela-Ascoli theorem to unifomly bounded sequence in $H^1$

Let $\{u_{i}\}$ be sequence of smooth function defined on $\Bbb{R}^n$ such that $\|u_i\|_{L^2(\Bbb{R}^n)}$ is uniformly bounded in $i$ and $\|\nabla u_{i} \|_{L^{^2}(\Bbb{R}^n)}$ is also uniformly ...
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### Finding metric projection mapping

Let $X$ = $c_0$, with supremum norm and $W$ = $\{ \{x_n\}_{n \geq 1} \in X$ $|$ $x_1=0\}$. Then how to find $P_W(x)$. Note that, $P_W(x)$ = $\{y_0 \in W : ||x-y_0|| = \inf ||x-y||,$ $y \in W \}$. In ...
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### unbounded self-adjoint operator approximation by bounded operators?

Consider an unbounded, self-adjoint operator $(T,D(T))$, on a separable Hilbert space $H$. We try to approximate $T$ by its truncation, that is, $T_{n}=P_{n}TP_{n}$, where $P_{n}$ is the projection ...
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### Bounded operator has closed image if and only if there exists positive a constant $c$ such that $c \| x \| \leq \| Tx \|$.

In my notes I have the following theorem If $T:X \to Y$ is a bounded operator where $X,Y$ are complex Hilbert spaces. Prove that $Im (T)$ is a closed set if and only if there exists positive constant ...
1 vote
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### Under what minimum assumptions a null set is necessarily meager?

$A\subset \Bbb{R}$ is a null set if $\lambda(A) =0$ ($\lambda$ :Lebesgue measure) $A\subset \Bbb{R}$ meager if $A$ is countable union of nowhere dense sets (sets whose closure contains no nonempty ...
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### Bounded from above convex function on a normed vector space is locally Lipschitz

Disclaimer: This thread is meant to record. See: SE blog: Answer own Question and MSE meta: Answer own Question. Anyway, it is written as problem. Have fun! :) I'm trying to generalize this result to ...
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### Exercise 4.19 (1) of Brezis

I am trying to solve the following exercise of Brezis' book on Functional Analysis. Let $(f_n)_{n \in \mathbb{N}}$ be a sequence in $L^p(\Omega)$ with $1 < p < \infty$ and let $f \in L^p(\Omega)$...
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### Orthogonal complement relationship [closed]

Suppose two spaces are orthogonal: $A\perp B$. Is $A^\perp \perp B^\perp$?
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### Does this implies $\mathcal{M}[T]_{\beta_1 \beta_2}$ is similar to the $\mathcal{M}[T]_{\beta_3 \beta_4}$?
$T\in\mathcal{L}(V)$ where $\dim(V) <\infty$ Consider $\beta_1, \beta_2, \beta_3, \beta_4$ four bases of $V$ . Does this implies the matrix $[T]_{\beta_1 }^{\beta_2}$ is similar to the matrix \$[T]...