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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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Adjoint of inverse map is the inverse of adjoint map?

Let $E,F$ be Banach normed spaces and $S,T \in L(E,F)$.Denote adjoint of $T$ as $T^* .$ Prove that if $ T^{-1}$ exist and $T(E)=F $, then $(T^{-1})^* = (T^*)^{-1}. $ Actually, I could not proceed beyond writing definition of adjoint map.
Esat Koç
  • 655
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Convergent sequence under linear extension (Hahn-Banach theorem)

In Functional Analysis, Hahn - Banach Theorem can be stated as follows: " Let $X$ be a real or complex vector spave and $p$ a real-valued functional on X which is additive, that is, for all $x$, $y$ in $X$, $$p(x+y) \le p(x) +…
Trần Linh
  • 725
4
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2 answers

Do bounded linear operators on a Banach space which are injective or have dense range form an open subspace?

We proved a theorem in our functional analysis class showing that the subspace of bijective bounded linear operators between two Banach spaces $X$ and $Y$ is open in the space $B(X,Y)$ of bounded linear operators. It was mentioned that the subspace…
adrija
  • 3,027
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2 answers

Separation in dual space

Let $X$ be a real Banach space and $X^*$ its dual space. Let $C^*$ be a weak$^*$ closed and convex subset in $X^*$ and $x^*\notin C^*$. Then there exists $x\in X$ such that $$ \langle x^*, x\rangle > \sup_{f\in C^*}\langle f, x\rangle. $$ I would…
blindman
  • 3,117
4
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Haar system forms an orthonormal system in $L_2[0,1]$

Haar wavelets are defined as: $$ \psi_{0,0}(t) = \begin{cases} 1, \text{ for } 0
Piotr Benedysiuk
  • 1,622
4
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Do spectrum change on a maximal sub algebra?

Can anyone help me to solve the following problem. We know if $B$ is a unital sub-algebra of a unital Banach algebra $A$ and if $a\in B$, then $\sigma _{A}[a] \subset \sigma _{B}[a] $(Here $\sigma _{T}$ denotes the spectrum with respect to the…
Timon
  • 2,401
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Approximations of functionals

The distance between a point and a set in a metric space or in a normed space is used in approximation of functionals (i read it in functional analysis book) can any body explain this please as i am unaware of approximation of functions
Prince Khan
  • 1,544
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Importance (applications) of functionals

Yes functional analysis is the study of functionals (up to some extend). Why we are so curious to study functionals? Some applications please for the motivations to me.
Prince Khan
  • 1,544
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sequence of bounded linear operators Tn: X to Y where X, Y are Banach spaces, ||Tn|| goes to infinity. Show there exists x ||Tnx|| goes to infinity

Suppose that $X,Y$ are Banach spaces. Let $T_n: X\to Y$ be linear operators for all $n\in \mathbb{N}$ such that $\lim_{n\to\infty} \|T_n\|=\infty$. Show that $\exists x_0\in X$ such that $\lim_{n\to \infty}\|T_nx_0\|=\infty$. I know that if…
Yang
  • 55
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Two versions of the Fredholm alternatives

Here are two version of the Fredholm alternatives among which I would like to know the relation: One version is from the appendix of Evans's Partial Differential Equations: Here $H$ is a Hilbert space. Another version is from an old post in…
user9464
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1 answer

Vectors converging to linearly independent vectors are eventually linearly independent

I hesitate if the following claim is true: Let $V$ be a normed vector space that is complete. For example, Hilbert space. And assume $\{v_1,...v_n\}$ is a subset of linearly independent vectors in $V$. Assume also that for any $v_k$ we have a…
user376810
  • 41
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Prove $T$ is surjective if $(Tx,x)\geq k(x,x)$ in Hilbert space

Suppose $H$ is a Hilbert space, and $T$ is a continuous operator satisfies $$(Tx,x)\geq k(x,x), k>0,x\in H$$ How to prove that $T$ is onto?
89085731
  • 7,614
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Let $H$ be hilbert and $T$ a BLO, such that $T:H\rightarrow H$. Prove that $\langle T(x),x \rangle = 0$ implies $T = 0$.

Let $H$ be hilbert and $T$ a BLO, such that $T:H\rightarrow H$. Prove that $\langle T(x),x \rangle = 0$ implies $T = 0$. Any hints to tackle this problem? i tried writing x as $x = u + v$ where $u \in Y$ and $v \in Y^T$ for some closed linear…
Kees Til
  • 1,958
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1 answer

Stone-Weierstrass theorem of $\mathbb{S}^2$

Someone told me that every continuous function on $\mathbb{S}^2$ could be expressed as a uniform limit of restrictions to $\mathbb{S}^2$ of polynomials. Does this result come from the Stone-Weierstrass theorem? Could anyone be able to explain to me…
user347575
4
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1 answer

Functional Analysis - Banach-Steinhaus theorem

How can I use the Banach-Steinhaus' Uniform boundedness principle in order to prove the following claim: If $x_n$ is a sequence of complex numbers such that the series $\sum_1^\infty x_n \chi_n$ converges for every sequence $ \chi_n \in l_p $ ($1…
joshua
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