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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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No closest point for a subspace $M = \{ \{x_n\} : \sum x_n 2^{-n} = 0\}$ of $c_0$

Let $M=\{\{ x_n \} \in c_0 : \sum_{n = 1}^\infty \frac{x_n}{2^n} = 0 \}$. Prove that $\forall x \in c_0 - M$, there is no closest point to $x$ in $M$. I want to solve the problem by proving that for every $y_0 \in M$, there exists $y_1 \in M$…
namasikanam
  • 515
3
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inequality for compact contained banach space

Let $X_0$,$X_1$ and $X_2$ be Banach spaces such that $$X_2\subset\subset X_1\subset X_0$$ I want to prove that for every $\epsilon>0$, there exists a constant $C_\epsilon>0$ such that for all $x\in X_2$, the inequality $$\lVert…
Emiya
  • 518
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Corollary of Uniform boundedness principle

Let $X$ be banach space, $Y$ be normed space, $\mathcal A \subset \mathcal B(X, Y)$ be some set of continuous linear operators $X \to Y$. I need to prove that if $$\forall x \in X, g \in Y^* \;\; \sup_{A \in \mathcal A} |g(Ax)| <…
dnes
  • 426
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Prove if Hilbert dimension is finite, then the Hilbert space as a vector space has the same dimension

This is from Kreyszig's functional analysis text, chapter 3.6 #2. The backward direction is easy and just the Gram Schmidt process on any basis. I am having some trouble on the forward direction. So we start with a totally orthonormal set M such…
Bill
  • 4,417
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3 answers

Question on the evaluation functional in $L^p$

I know that for $1 \leq p < \infty$ we have $C_c(X)$ dense in $L^p(X,\mu)$. In $L^p(X,\mu)$ the evaluation functional is not bounded, but in $C_c(X)$ it is. So if I define $T_x : C_c(X) \to \mathbb{R}$ as $T_x f = f(x)$ for $x \in X$ I can extend…
user8469759
  • 5,285
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What is the Hilbert transform of 1?

I am studying singular integrals and I have understood that if we have an operator $T$ which is bounded in $L^2$, then we can extend this operator for functions in $L^{\infty}$ to BMO space. This is the case of the Hilbert…
pitàgores
  • 181
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A bounded linear operator from $X ^*\rightarrow Y^*$ is weak*-weak* continuous iff It is the adjoint of some Bounded linear operator $X\rightarrow Y$

(this question is not a duplicate of this one since the latter only addresses the situation in the case of Banach spaces) Let $X,Y$ be normed vector spaces and $B:Y^*\rightarrow X^*$ a linear operator. We want to show that $B$ is $weak*-weak*$…
Muselive
  • 948
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Inequality used to prove the lemma Brezis-Lieb

How do I prove the inequality below? $$\left|\left|a+b\right|^p-\left|a\right|^p-\left|b\right|^p\right|\leq C~(\left|a\right|^{p-1}\left|b\right|+\left|a\right|\left|b\right|^{p-1} ), \ \forall \ a,b \in \mathbb R$$ where $C$ is a constant that…
César
  • 31
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Is any subset of a $A$ of a separable norm linear space $X$ is separable?

I just read the theorem "If X is norm linear space and if $X^*$ is separable $X$ is also separable" from Bachman and Narichi, page 201. To prove it they consider the set $S=\{f \in X^* :\lVert f\rVert=1\}$. And the writer said that $S$ is…
Andy
  • 2,246
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Any positive distribution is a measure.

Is it true that the limit of positive distributions is still a positive distribution? I ask this, because I read that every positive distribution is a measure. So I would like to prove that if $(T_n)$ sequence of positive distributions such that…
eraldcoil
  • 3,508
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2 answers

how to show that $Y= \{f\in L^{2}[0,1] \mid f(x)\geq x \text{ a.e.}\}$ is weakly closed in $L^{2}$

Problem: Let $Y= \{f\in L^{2}[0,1] \mid f(x)\geq x \text{ a.e.}\}$. Show that $Y$ is weakly closed in $L^{2}$. My thought about solving this problem is that consider a sequence $\{f_{n}\}$ which converges to some $f$ weakly and to show that $f$…
Brain Zhang
  • 689
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1 answer

Definition of the second order functional derivative

In Appendix A of Density Functional Theory An Advanced Course by Eberhard Engel, Reiner M. Dreizler functional derivative is introduced by studying the Taylor expansion of some functional $F$ around function $f$. $F$ is treated as a function of…
cHewap
  • 93
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2 answers

Is a linear map of norm $1$ always an isometry?

Let $(E,\|.\|_{E})$ and $(F,\|.\|_{F})$ be two normed spaces and let $f:E\longrightarrow F$ a linear map such that $\|f\|=1$. I don't know if this means that $f$ is an isometry $(\|f(x)\|_{F}=\|x\|_{E}, \,\, \forall x\in E)$. Thanks
Serges
  • 479
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Why is $ \|T(x) \|\le \| T\| \|x \|$?

Let $X$ and $Y$ be normed linear spaces and let $T$ be a bounded linear operator from $X$ to $Y$. The norm of $T$ is defined as $$\|T \|=\sup\{\|T(x) \|\;:\;\|x \|\le 1\}. $$ From the definition of the norm, we can say that $\|T(x) \|\le \|T…
Bhavish Suarez
  • 684
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1 answer

Finite products of topological vector spaces

A finite product of an infinite Hilbert space is isometric to itself, since the isometry class of a Hilbert space is determined only by the cardinality of its dimension. But what about less 'nice' topological vector spaces of infinite dimension? I…
Constant
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