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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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The norm of conjugate operator in Banach space

I want to prove that $ \begin {Vmatrix} T \end {Vmatrix} = \begin {Vmatrix} T^{*} \end {Vmatrix}$ where $ T: E_1 \to E_2$, $T^{*}: E^{*}_2 \to E^{*}_1$ are operators between two Banach spaces and its dual spaces respectively. Let $f \in E^{*}_2$,…
Invincible
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Why is a set complete in $C[0,1]$ also complete in $L_2[0,1]$?

I was asked to prove that $\{{t^{3n}}\}_{n=0}^{\infty}$ is complete in $L_2[0,1]$ (complete system). And the solution says that it is sufficient to show completeness in $C[0,1]$. Why is that? Is it simply because continuity on a closed interval…
Meitar
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Metric in Hilbert Space defined as $d := d(x, y) = \inf_{\lambda} \|\ x + \lambda y \|\ $, where $\|\ y \|\ = 1 $

Question: Let $d := d(x, [[y]]) = \inf_{\lambda} \|\ x + \lambda y \|\ $, where $y$ is a unit vector; and $[[y]]$ denotes the span of vectors (with norm of 1); show that: (a) $d = \|\ x + \lambda_0 y \|\ $ for some $\lambda_0$ , (b) $| \langle x,y…
Dragonite
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Question about dirac Delta function

Let $\delta(x)$ denote a Dirac Delta function on $R$. Then, I know that $$\delta(x) = 0 \text{ if } x \neq 0$$ and $$\int_R \delta(x) f(x) dx = f(0)$$ Then, since $\delta(x)$ is zero everywhere except at $0$, it seems to me that $$\int_{(-\epsilon,…
nan
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Basic question about superposition operator

Let $\Omega$ be an open set in $\mathbb{R}^n$, $n\geq 1$. If $H$ and $Z$ are Hilbert spaces and $T:\Omega \times H \to Z$ is an operator, suppose that for all $\omega \in \Omega_s \subset \Omega$ and all $h \in H$, we have $$T(\omega, h) = 0.$$ If…
Glaos
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Generalisation of Schauder fixed point theorem

My problem is regarding Theorem 3 in Chapter 9 of McOwen's PDE text. The Schauder fixed point theorem states that: Let $X$ be a real Banach space. Suppose $A \subset X$ is compact and convex, and assume also $T:A \rightarrow A$ is continuous. Then…
anegligibleperson
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If two norms on an infinite dimentional vector space generate the same completion (generates two isomorphic banach spaces) are they equivalent?

If two norms on an infinite dimensional vector space X make the same completion(the clousure of the vector space with respect to two norms are the same ) are they equivalent norm ? If not would you give me a counter example ? Thanks in advance
89701407
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If $T$ is a bounded linear operator between Hilbert Spaces and $\lVert{T}\rVert = \lVert{T^{-1}}\rVert =1$, is $T$ unitary?

If $T:K \rightarrow L$ is a bounded linear operator between two Hilbert Spaces $K$ and $L$, then we have automatically that if $T$ is unitary, then $\lVert{T}\rVert = \lVert{T^{-1}}\rVert = 1$ by the following: $\lVert{Tx}\rVert^{2}_{L} = \langle…
sc636
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stuck on problem on completeness of subspace of C[0,1]

I am really stuck in the following problem: Let $X=C[0,1]$ with the inner product $\langle x,y\rangle=\int_0^1 x(t)\overline y(t)\,dt$ $\forall$ $x(t),y(t)\in C[0,1]$ $X_0 =\{x(t) \in X :\int_0^1 t^2x(t)\,dt=0\}$and $X_0^\bot$ be the orthogonal…
Abhishek Shrivastava
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seminorm & Minkowski Functional

It is known that if $p$ is a seminorm on a real vector space $X$, then the set $A= \{x\in X: p(x)<1\}$ is convex, balanced, and absorbing. I tried to prove that the Minkowski functional $u_A$ of $A$ coincides with the seminorm $p$. Im interested on…
Juniven Acapulco
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Is the closure of the sobolev space $H^1(\Omega)$ equal to $L^2(\Omega)$?

I am not mathematician (engineer) and would like to know whether the closure of the well-known sobolev space $H^1(\Omega)$ is equal to $L^2(\Omega)$. My attempt: $H^1(\Omega)$ is dense in $L^2(\Omega)$ (because $C_0^\infty$ is contained in…
Abdulrahman Obaid
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Pre-requisites for Riesz-Nagy

Can anyone tell me what the reader must already know in order to meaningfully read "Functional Analysis" by F Riesz and Bela Sz Nagy from the start to the very end?
Ishihara
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$U$ open neighbourhood of the origin. Then there is an open neighbourhood $N$ of the origin st $\alpha N \subset U$

Let $(V,T)$ be a topological vector space over $\mathbb{F}$ where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$. Claim: Let $U$ be an open neighbourhood of the origin. Then there is an open neighbourhood $N$ of the origin s.t $ \ \alpha N…
Olba12
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Problem regarding Annihilator of a matrix and the minimal polynomial.

I have some doubt regarding annihilator of an matrix. Please help me to understand the subject. Let $T$ be a n by n Complex matrix. Let $m(x)$ be the minimal polynomial for $T$. Now suppose $f(T)=0$ for some $f$-holomorphic function on a…
Timon
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Show that $\|(\lambda - T)^{-1}\| \leq 1/(1-|\lambda|)$

Let $X$ be a Banach space and let $T \in L(X)$ satisfy $\|Tx\| =\|x\|$ for each $x \in X$. Suppose that the range of $T \neq X$ and let $0 < |\lambda| < 1$. Assuming that $λ \in \rho(T)$, show that $\|(λ − T)^{−1}\| \leq 1/(1−|λ|)$. Here $\rho(T)$…
ronalddb89
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