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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).

Functional analysis is 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, including spectral theory, as well as measure, integration, probability on infinite dimensions, and also manifolds with local structure modeled by these vector spaces.

For basic questions about functions use more suitable tags like , or .

52582 questions
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Question on equality with mollifier

So I have this homework. Note that $J_\epsilon$ is the standard mollifier. If $\Omega\subset\mathbb{R}^n$ open and $u\in C(\Omega)$, show that $J_\epsilon * u\rightarrow u$ uniformly on every compact subset of $\Omega$ as $\epsilon\rightarrow0$. At…
rom
  • 831
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M*f(x) is not continuous

Given $f: R^n\to R $ an integrable function. Then the centered maximal operator is defined by $$M^{*}f(x)= \sup_{r>0} \dfrac{1}{\mu(B(x,r))} \int_{B(x,r)}|f(y)|\,d\mu(y) .$$ Here the supremum is taken over all the balls in $\mathbb{R}^{n}$…
majduleen zeyadeh
  • 73
3
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1 answer

$l^p$-Product Norm of Banach spaces

Let $\|(x,y)\| _{X\times Y} := \sqrt{\|x\|_X^2+\|y\|_Y^2}$ be the $l^2$ product norm of the banach spaces $X,Y$. I somehow struggle with proving that this norm satisfies the triangle inequality. EDIT: Using the TI of the norms in $X$ and $Y$ I…
EpsilonDelta
  • 2,079
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Separable $X^* \implies$ bounded $\{x_n\} \in X$ have subsequence st. $\lim_{k \to \infty}f(x_{n_k})$ exists $\forall f \in X^*$

In full: Show that if $X^*$ is separable then any bounded sequence $(x_n) \in X$ has a subsequence $x_{n_k}$ such that $\lim_{k \to \infty}f(x_{n_k})$ converges for every $f \in X^*$ What I have so far: let $\{f_i\}$ be a countable dense subset of…
Christian Fieldhouse
  • 879
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Weakly convergent $\{x^{(n)}\} \in \ell^1$ bounded

An exam question asks: Show that if $$x^{(n)} = (x^{(n)}_1,x^{(n)}_2,x^{(n)}_3,...)$$ is a sequence in $\ell^1$ that converges weakly in $\ell^1$ to some $x \in \ell^1$ then $x^{(n)}_j \to x_j$ for all $j$ and $||x^{(n)}||_{\ell^1}$ is bounded…
Christian Fieldhouse
  • 879
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Why is the dual exponent what it is?

The dual $p'$ of an exponent $p \in [0, \infty]$ is: $$p'= \begin{cases} \frac{p}{p-1}, & 1 < p < \infty \\ \infty, & p=1 \\ 1, & p= \infty. \end{cases}$$ Why is this the dual of an exponent?
mavavilj
  • 7,270
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1 answer

In a $C^*$-algebra, why is the norm of a positive element attained by a positive, linear functional of norm 1?

By Hahn-Banach it holds in any Banach space that the norm of a positive element attained by a functional of norm $1$. But why is it true in a $C^*$-algebra that norm of any positive element is attained by a positive, linear functional of norm $1$?
Gaute Schwartz
  • 171
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Showing the closed ball of the dual space X* is weak-star closed

I am practicing some elementary problems from functional analysis. I am trying to show when X is a normed space then the closed unit ball of its dual $ X^* $ is weak-star closed. I can show whenever a sequence weak-star converges in the closed…
Carey Washington
  • 51
3
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Find norm of linear operator $Tf(t)=f(t)+\int_0^t(1-2s)f(s)ds$

Let $T:C[0,1]\to C[0,1]$ be linear operator defined as follows: $Tf(t)=f(t)+\int_0^t(1-2s)f(s)ds$. Find its norm where norm is is supremum norm. I am not sure how to do it. It is easy to see that when we take $t\in[0,1/2)$, then we get best result…
SekstusEmpiryk
  • 2,244
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Fourier coefficients in $L_1$

Let $f∈L_1[0,\pi]$. Consider the Fourier coefficient $\{c_n\}$ in trigonometric system $\{\sin(nx)\}$. 1)Is it necessary to converge series $\sum_{n=1}^\infty |с_n|$? 2)Under what conditions does the series is converge $\sum_{n=1}^\infty c_n^2$ As…
user1223
  • 197
3
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1 answer

Commutant of two bounded linear operators

Let $E$ be an infinite-dimensional complex Hilbert space and $A,B\in \mathcal{L}(E)$. Why it is impossible to find $c\in \mathbb{C}^*$ such that $[A,B]=cI$?
Student
  • 4,914
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Prove that for every compact $K \subset \{z\in\mathbb{C}:|z|<1\}$ there is operator $T$ on Hilbert space such that $\sigma(T) = \sigma_p(T) = K$

Possible Duplicate: Compact sets as point spectrum of a bounded operator Prove that for every compact set $K \subset \mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ there is operator $T$ on Hilbert space such that $\sigma(T) = \sigma_p(T) = K$ where…
Mikhail Gozhev
  • 111
3
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3 answers

Fixed point in a continuous map

Possible Duplicate: Periodic orbits Suppose that $f$ is a continuous map from $\mathbb R$ to $\mathbb R$, which satisfies $f(f(x)) = x$ for each $x \in \mathbb{R}$. Does $f$ necessarily have a fixed point?
salar_ve
  • 49
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Application of the Hahn- Banach theorem

Let $E$ be a normed space and $F$ be a subspace of $E$. Show that $F$ is dense in $E$ if and only if all the linear and continuous functional on $E$ satisfying $f\vert _F=0 $ are identically zero ($f = 0$).
daisy
  • 325
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2 answers

$ \left( ||T^n||\right)^\frac1n \leq ||T|| $ as $n\to\infty$?

Given an operator $T\in B(X,X)$, I'm trying to prove that $$\lim_{n\to\infty} \left( ||T^n||\right)^\frac1n \leq ||T|| $$ I can show that $||T^n||\leq||T||^n$, but only for finite $n$. How can I be sure that this holds in the limit? I'm a bit…
Belen
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