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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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Is this set dense in $L^2(0,1)$?

What I have here is the neighborhood $U := \{z\in\mathbb C : |z| < \varepsilon\}$ of zero in $\mathbb C$ and continuous functions $a,b : U\times [0,1]\to\mathbb C$ such that $|za(t) + b(t)|\ge\delta > 0$ for all $(z,t)\in U\times [0,1]$. My…
amsmath
  • 10,633
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$L^p$ norm is lower semi continuous but not continuous?

It's written in my course of functional analysis that $L^p$ norms is lower-semicontinuous but not continuous. For me, continuity of $\Phi: L^p\to \mathbb R$ is : if $f_n\to f$ in $L^p$ then $\Phi(f_n)\to \Phi(f)$. For $L^p$ space, it looks…
user657324
  • 1,863
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1 answer

Banach algebra on $C^*$-algebra

Let $A$ be a $C^*$-algebra, and let $a$ in $A$ be normal, and let $B$ be the $C^*$-algebra generated by $a$. Suppose that $f:\sigma(a)\to\mathbb{C}$ is continuous. Show that there exists an element $x$ in $B$ such that $\Phi(x)(s)=f(\Phi(a)(s))$ for…
Simon Tian
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In a topological vector space , every neighborhood of $x$ can be write as the union of the local base of $x$ .

In Rudin's functional analysis , let $X$ denote a topological vector space . He states that the open sets of $X$ are precisely those that are unions of translates of members of the local base of $0$ . Since for any nonempt subset $E$ , we can find…
J.Guo
  • 1,627
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Compute the norm of $\phi$

We have , $E=C([0,1],\mathbb{R})$ equipped with norm $\left \| . \right \|_\infty $ and $\phi:E\rightarrow E$ $\phi(f)(x)=\int_{0}^{x}tf(t)dt+x\int_{x}^{1}f(t)dt$, for any $x\in [0,1]$ I proved that $\phi$ is linear and continuous (Lipschitz )…
Pedro Alvarès
  • 720
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Finding an Orthogonal Complement of a Subspace

Let $C([0,1])$ be the space of all continuous function on $[0,1]$. Define the an inner product in the space: $$ \langle f,g \rangle = \int_0^{1}f(t)g(t)\;dt $$ Let $K$ be a subspace of $C$: $$ K=\{ax+bx^2;a,b \in \mathbb{R}\} \subset C $$ How do I…
A Slow Learner
  • 1,201
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3 answers

Diameters, distances and contraction mappings on a subset of $C_{\mathbb{R}}[0,1]$

Let $$M=\{f\in C_{\mathbb{R}}([0,1]): f(0)=0\le f(t)\le f(1)=1,\text{ for }t\in [0,1]\}$$ where $C_{\mathbb{R}}([0,1])=\{f:[0,1]\to \mathbb{R}:f\text{ is continuous on }[0,1]\}$ is Banach space with norm $\|f\|_\infty=\sup \{|f(t):t\in [0,1]\}$ .…
Inverse Problem
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functional calculus $\theta(1/f)=\theta(f)^{-1}$

If $T\in B(H)$ is normal,and if $f\in C(S_p(T))$ is never zero,how to prove that the functional calculus $\theta$ for T satisfies $$\theta(1/f)=\theta(f)^{-1}$$
89085731
  • 7,614
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Are there $L^2$ functions on the boundary of the disk that are not in the image of the given extension?

Consider the Sobolev space $W^1$ that is the closure of $\mathcal{C}^{\infty}(\bar{D_1})$ with respect to the norm $$|\phi|_1^2=|\phi|^2_{L^2(\bar{D_1)}}+|\nabla \phi|^2_{L^2(\bar{D_1)}}.$$ I have proven that one can continuously extend the map…
Anri Rembeci
  • 289
2
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2 answers

Preduals and $c_0$

I know that $c_0$ does not have a predual, but if we put an equivalent norm on $c_0$, can this space have a predual?
user124910
  • 3,007
2
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2 answers

Bounded linear operator between banach spaces with dense image

If X,Y Banach spaces and T: X $\to$ Y bounded linear operator with T(X) dense and not equal to Y. Prove that there exists y in Y such that $\left\lVert{x_n}\right\lVert \to \infty$ for every sequence $x_n$ for which $Tx_n\to y$. I know I am…
Plom
  • 661
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1 answer

Implications of continuity

A mapping $T:X\rightarrow Y$ between metric spaces $(X.d),(Y,\hat d)$ is continuous at $x_0$ if for every $\epsilon>0$ there is a $\delta>0$ such that $\hat d(Tx,Tx_0)<\epsilon$ for all $x$ satisfying $d(x,x_0)<\delta$. Consider such a mapping $T$,…
cangrejo
  • 1,279
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1 answer

$sup_{f\in X^* , ||f||\le1}\sum_{n=1}^N |f(x_n)| = max _{\epsilon_n \in \{\pm1\}}||\sum_{n=1}^N \epsilon_nx_n ||$

Let $\{x_n\}_{n=1}^N $ be elements in a normed space $X$. I want to prove that $$sup_{f\in X^* , ||f||\le1}\sum_{n=1}^N |f(x_n)| = max _{\epsilon_n \in \{\pm1\}}||\sum_{n=1}^N \epsilon_nx_n|| $$ im not sure how. I thought it may be related to the…
user335501
2
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1 answer

Is this restriction a topological isomorphism?

Let $(E,\tau_1)$ and $(F,\tau_2)$ be Hausdorff locally convex topological vector spaces with topologies $\tau_1$ and $\tau_2$, respectively. Let $T:(E,\tau_1)\longrightarrow (F,\tau_2)$ be a topological isomorphism, that is, $T$ is linear, bijective…
serenus
  • 851
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Prove: the inverse of an absolutely continuous operator $A:E_1 \to E_2$ need not be continuous.

I'm studying about statistical learning theory and I bumped into the following statement in my study material: Claim: If a linear normed space $E_1$ contains bounded noncompact sets, then the inverse operator $A^{-1}$ for an absolutely …
jjepsuomi
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