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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 .

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Dimension of the range of an operator and its adjoint

Proof for $\dim(R(T))=\dim(R(T^{*}))$ for a linear operator in a Hilbert space. $T$ is the operator and $T^{*}$ is its adjoint. I would like to know about the authenticity of the following line of proof of the above fact and get directed to a…
Hamza
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Two linear functionals are equal

Let, $f$ and $g$ be two linear functionals such that ker$f$=ker $g$ and $f(a)$=$g(a)$. Then to prove $f(x)$=$g(x)$.
Topology
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Motivation behind the definition of topological vector space

I see in a book the following example as a motivation to define topological vector space. $$+: \Bbb R^n \times \Bbb R^n \to \Bbb R^n$$ defined by $(x,y) \to x+y$ and $$.: \Bbb R^n \to \Bbb R^n$$ defined by $x \to \alpha x$ where $\alpha$ is a…
Anonymous
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Is every Banach space isometrically isomorphic to the dual of a normed space?

I have the following question: Given a Banach space $V$ over the field $\mathbb{K}$, is it true that there is always a normed space $W$ over the field $\mathbb{K}$, such that $V$ and $W'=\mathscr{L}(W,\mathbb{K})$ are isometrically isomorphic (with…
Monkey D Ruffy
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Completion of a normed space with respect to equivalent norms

Suppose we have two normed linear space $(X,\|\cdot\|_1)$ and $(X,\|\cdot\|_2)$. Also, the norms are equivalent to each other. Do we have same completion of $X$ with respect to $1$ and $2$ norms? In other words, do we have an isometric isomorphism…
Neon
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Completion of pre hilbert space

Let $L$ be a Hilbert space and $T$ be a linear densely defined operator, $T: D(T)\subset L \to L$ , $\overline {D(T)}=L$ We can make $D(T)$ , a prehilbert space by defining an inner product $\langle \mathbf{x},\mathbf{y}\rangle_T=\langle…
Neon
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Contraction mapping problem

Let $T$ be the following operator on $C[0,1]$: $$(Tu)(t) = u(0) + \lambda\int_0^t u(\tau)d\tau$$ where $\lambda \in (-1,1) \subset \mathbb{R}$. Then I need to show $T$ is a contraction. So I need $$||Tu - Tv|| \leq…
Nick
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Bessel sequence and synthesis operator

A sequence $(f_{k})_{k\in \mathbb{N}}$ is called a Bessel sequence in a Hilbert space $H$, if there exists $B>0$ such that $$\sum_{k\in \mathbb{N}}|\langle f,f_{k}\rangle|^{2}\leq B\|f\|^{2}$$ for all $f\in H$. The operator $$T:l^2 \rightarrow H, …
user161913
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Integral over compact set and open set

Let $D \subset \mathbb{R}^d$ be open and let $a_{ij}\in L_{loc}^{1}(D\,;\mu),\,a_{ij}=a_{ji},\,1\leq i,j \leq d$ ($\mu$ is Lebesgue measure on $D$). We define $\mathcal{S}:C_{0}^{\infty}(D) \times C_{0}^{\infty}(D)\to \mathbb{R}$ by…
ko4
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every compact subset of a TVS is bounded

I'm self-studying Functional Analysis. The following is Exercise 4.2.4 of Conway's Functional Analysis. Let $X$ be a topological vector space. Show that every compact subset of $X$ is bounded. For this I just know every open cover of compact set $K$…
niki
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Let $u_n, u \in L^2$. If $\int u_nv \to \int uv$ for all $v \in H^1$, does $\int_{}u_nh \to \int_{}uh$ for all $h \in L^2$?

Let $\Omega$ be a bounded domain and let $u_n$ and $u \in L^2(\Omega)$. Question: If $\int_{\Omega}u_nv \to \int_{\Omega}uv$ for all $v \in H^1(\Omega)$, does $\int_{\Omega}u_nh \to \int_{\Omega}uh$ for all $h \in L^2(\Omega)$? I think so. Let $v_m…
delimit
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Reverse Uniform Boundedness Principle

I believe that what I am about to ask has a negative answer, but I can't seem to find a quick counterexample. Consider a family of bounded operators $\{T_\alpha\}_{\alpha \in \mathcal{A}} \subset \mathcal{B}(X)$ where $X$ is a Banach space. The…
Beni Bogosel
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If $T ( x,y ) = 2x + y , ∀( x,y )∈ \mathbb{R}^2$. Determine $||T||$.

Define $T : \mathbb{R}^2 → \mathbb{R}$ ( $\mathbb{R}^2$ & $\mathbb{R}$ being equipped with the Euclidean norm) by $T ( x,y ) = 2x + y , ∀( x,y )∈ \mathbb{R}^2$. Determine $||T||$. My thoughts:- We know that $ \qquad \left\|{T}\right\| = \sup…
ghugni
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Convergence, Banach-Steinhaus

Let $(\phi_n)$ be a sequence of continuous linear functionals from $X$ - Banach space to $\mathbb{R}$ such that $(\phi_n(x))$ is convergent for all $x\in X$. Show that if a sequence $(x_n)$ is convergent in X norm, then $(\phi_n(x_n))$ is…
luka5z
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Does closed image imply the adjoint has weak^* closed image?

Question: If a bounded operator between Banach spaces has closed image, does its adjoint necessarily have weak$^*$-closed image? The equivalence of these two conditions is claimed in Theorem A.3.48 of Dales' "Banach algebras and automatic…
Zsolt Tanko
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