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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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Does a sequence bounded by a function of $L^{\infty}$, converges in $L^{\infty}$?

I want to know if this statement is right? Let $(g_n)$ be a measurable sequence from $[0,T]$ to $\mathbb{R}^n$ such that $\exists \beta \in L^{\infty}_{\mathbb{R}_+}([0,T])$ where $|g_n(t)| \leq \beta(t) \forall t \in [0,T]$ then $(g_n) \in …
Myriam
  • 105
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Find a function which satisfies an integral equation

How can I find a function, $$ f: \mathbb{R} \to \mathbb{R} $$ which satisfies the following equation: $$\cos\left(t^2\right) = \int_{-\infty}^{\infty} e^{itx}f(x)\,dx$$
Kerry
  • 795
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2 answers

About closedness and boundedness of $H=\left\{(x_n)\in \ell^2(\mathbb{N})\mid\sum \frac{x_n}{n}=1\right\}$

Let $H=\left\{(x_n)\in \ell^2(\mathbb{N})\mid\sum \frac{x_n}{n}=1\right\}$. To check which one is true: (a) $H$ is bounded (b) $H$ is closed (c) $H$ is a subspace (d) $H$ has interior points My try: (c) is not true as $x_n\in H$ does not imply…
Learnmore
  • 31,062
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Definition of space $L^2(\mu)$ where $\mu$ is a Borel probability measure on $\mathbb R$.

Let $\mu$ be a Borel probability measure on $\mathbb R$ with compact support. Consider the space $L^2(\mu)$. It is the first time that I meet this space (usually I have $L^2(\mathbb R)$). Is it still a Banach space? Is it still a Hilbert space? By…
Lely
  • 1,057
3
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2 answers

Two contradictory results on unbounded linear operators?

I have known that if $A$ is a densely defined (unbounded) operator with domain $D(A)$ such that $\langle Ax,x\rangle=0$ for all $x\in D(A)$, then this does imply that $Ax=0$ for all $x\in D(A)$. This result may be found e.g. in Schmudgen's new book…
Math
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Weak convergence, $L^{2}$

I have a question about weak convergence. Let $(S,\Sigma,m)$ be a measure space. $(f_{t})_{t>0}$ be a family of square integrable functions. (i.e. for every $t>0$, $f_{t} \in L^{2}(S;m)$) and $f$ be a square integrable function. Question Let…
sharpe
  • 928
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Consider $c_{00}$ as a subspace of $(\ell^p,\|\cdot\|_p)$. Show that the closure of $(c_{00},\|\cdot\|_1)$ is $\ell^1$

Consider $c_{00}$ as a subspace of $(\ell^p,\|\cdot\|_p)$. Show that the closure of $(c_{00},\|\cdot\|_1)$ is $\ell^1$, closure of $(c_{00},\|\cdot\|_2)$ is $\ell^2$ and closure of $(c_{00},\|\cdot\|_\infty)$ is…
tattwamasi amrutam
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Compact operator in Hilbert Space

$H$ is a Hilbert space and $A$ is a bounded operator on $H$. If $A^*A$ is compact, is it necessarily that $A$ is compact?
Summer
  • 6,893
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2 answers

What is $H_0^1$ space?

I'm reading a book and it says that the $H_0^1(\Omega)$ space is defined as "the completion of $C_0^\infty(\Omega)$ w.r.t the Sobolev norm $\| \cdot \|_1$, where $C_0^\infty(\Omega)$ is the space of infinitely differentiable functions which are…
Ana Uspekova
  • 529
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1 answer

Is it incorrect to say that a functional "maps functions to numbers"?

Does a "functional" always takes in a function and spit out a number? This is what a professor said in class a long time ago but now I am studying Frechet derivative and a claim was made that a linear functional is $Ax: V \to W$. Can $W$ be…
Shamisen Expert
  • 8,092
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3 answers

An equality in Hilbert spaces

To understand a proof in functional analysis I need to understand why the following equation is true: $$\lVert x\rVert^2 - \sum_{j=1}^n |x_i|^2 = \Biggl\lVert x-\sum_{i=1}^nx_ie_i\Biggr\rVert^2$$ Where $x\in H$ ($H$ a Hilbert space) and $x_i=…
Duke
  • 529
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2 answers

The spectrum of a bounded linear operator

Suppose $X$ is a Banach space. For $T\in L(X,X)$, let its spectrum be $\sigma(T)$. Show that $\lambda\in\sigma(T)\Rightarrow\lambda^{n}\in\sigma(T^{n}),\ \forall n\in\mathbb{N}$. Show that the converse is true for $\mathbb{C}$ but not for…
user16859
  • 705
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0 answers

Hahn-Banach from "Every vector space has basis"

What is the simplest way to prove Hahn-Banach starting from the AC-equivalent that every vector space admits a basis?
Damian Reding
  • 8,773
3
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1 answer

Is there a sequence of continous function pointwise convergent to Riemann function?

I'm reading Baire's Category theory recently. One can find the following theorem in Chapter 4 of Stein's Functional Analysis: "Suppose that $\{f_n\}$ is a sequence of continuous complex-valued functions on a complete metric space $X$, and $\{f_n\}$ …
AG learner
  • 4,523
3
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1 answer

This operator has to be self-adjoint

Let $T$ be a self-adjoint strict contraction ($||Tx||<||x||$ for all $x\in H\setminus\{0\}$ where $H$ is a Hilbert space). One observe that the expression $T\sqrt{I-T^2}^{-1}$ is well-defined on the range of $\sqrt{(I-T^2)}^{-1}$. Can someone help…
Luther Baker
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