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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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Prove $(S+T)^\times = S^\times +T^\times$.

$T^\times$ and $S^\times$ are the adjoint operators of $T,S\in B(X,Y)$, $X$ and $Y$ normed spaces. $T^\times$ and $S^\times$ are defined on the dual spaces which contain the ranges of $T$ and $S$, respectively. Prove $(S+T)^\times = S^\times…
Desperate Fluffy
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Computing adjoint of a linear operator

I would like to know how to find an adjoint of an operator $T$ on a Hilbert space. I tried to find out on my own but it's not solid. Here is what I did: I picked a concrete example. Let $H=\ell^2$ and let $R: H \to H$ be the right shift operator.…
user167889
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Weak convergence in Banach spaces

In Rudin's book <> Page 66, it says "If $X$ is a infinite dimensional topology vector space, then $X$ under the weak topology is not locally bounded" . Hence I think the topology of any (infinite)Banach space $X$ is different from the weak topology,…
Shijie Dong
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$X=C[0,1]$ is a Banach space, $M=\{f\in X: f(0)=0\}$, prove $M$ is closed, find explicit formula for the quotient norm, and find an isomorphism.

Here is my question: Let $X$ be a Banach space $C[0,1]$ with the supremum norm. Let $M=\{f\in X: f(0)=0\}$. Show that $M$ is closed. Find an explicit formula for the quotient norm $\|[f]\|$ for $[f]\in X/M$. Find an isometric isomorphism from…
user3784030
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If $\sum_{n=1}^\infty \|x_n\|\lt\infty$, , then $\lim_{k\to\infty}\sum_{n=1}^k x_n$ exists

Here is my question: Let $X$ be a Banach space with norm $\|·\|$. Prove that, for any sequence $\{x_n\}$ in $X$, if $\sum_{n=1}^\infty \|x_n\|\lt\infty$, then $\lim_{k\to\infty}\sum_{n=1}^k x_n$ exists. Here is what I got: Given that…
user3784030
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Bump Functions on Open Intervals

I just have a quick question about bump functions. If we're dealing with $C^{\infty}_{0}((0,\infty))$, i.e. all smooth bump functions on $(0,\infty)$, obviously any $f\in C^{\infty}_{0}((0,\infty))$ vanishes at infinity, but does it also have to…
TinaBelcher
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$L^{2}$ convergence, bounded function.

Let $X$ be a metric space and $\mathcal{B}(X)$ be a Borel $\sigma$-algebra on $X$ and $\mu$ be a finite measure on $X$. We consider continuous functions (denoted by $\{f_{n}\}$) on $X$. If $f_{n}\to g$ in $L^{2}(X\,;\mu)$ and $g$ is bounded on $X$,…
ko4
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Bounded sequence in L1 ,counterexample

I'm looking for the simpliest counterexample, that bounded sequences in $L^1(\Omega)$ with $|\Omega|<\infty$ may not have weakly convergent subsequence. I'd appreciate if you could at least give me any reference where I can find it. Thanks in…
KKK
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Notion of simplicity of a function(al)

Given a function (functional actually) $f(x,g(x))$, can a notion of simplicity be attached with respect to the function $g(x)$? (all functions and args are real). Specifically, intuitively one could say that the function $f(x,0)$ is simpler than…
Jorge
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What can we say about operators in Lp that commute with translation?

Suppose I have an operator $T: L^p(\mathbb{R}^d) \rightarrow L^r(\mathbb{R}^d)$ that commutes with translation: $\tau_h \circ T = T \circ \tau_h$. Can I conclude that $T$ is a convolution? If not, is there anything else I can say? For $L^1$ and…
James Fennell
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Continuous Linear Operator

Consider the two linear spaces: $l^{2} = \left\{x = (x_1, x_2, . . . ) : \sum_{k=1}^{\infty} |x_k|^{2} < \infty\right\}$ with norm $||x||_{2} = (\sum_{k=1}^{\infty} |x_k|^{2})^{\frac{1}{2}}$, and $l^{\infty} = \left\{x = (x_1, x_2, . . . ) :…
ghjk
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Sum of subspaces is closed => the subspaces were closed?

I have the following question: We know that if $A$ is a closed subspace of a Hilbert space $\mathcal{H}$ and $B$ is a finite dimensional subspace such that $A\cap B=\{0\}$, then $A\dot{+}B$ is necessarily closed. Does this also go the other way…
heini
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How does Bessel's inequality imply convergence?

I lecture today we talked about Bessel's inequality and it's use in showing convergence of orthonormal sequences in a Hilbert space. $\sum_{n=1}^\infty \lvert \rvert^2 \leqslant \lVert x\rVert^2 \implies e_n\to 0$ (weakly) I am not sure how…
user3784030
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Dense domain of an operator

Suppose that $T$ is a (possible unbounded) self-adjoint operator on a Hilbert space $H$, thus the domain $D(T)$ of $T$ is dense in $H$ and the graph of $T$ is closed in $H\times H$. I want to prove that $D(T^2):=\{x\in D(T):Tx\in D(T)\}$ is dense in…
MuHo33
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Minima of convex Gateaux differentiable maps

I conjectured that: if $E$ is a reflexive Banach space and $F: E \to \mathbb{R}$ a convex Gateaux differentiable map (in other words all the directional derivatives $\frac{\partial F}{\partial \xi}(u)$ exists continuous in $u$ and linear in $\xi$)…
Antonio Alfieri
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