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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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Polynomial and spectrum of a Operator .

I need help with the following question : Let us consider $X$ to be a $\mathbb C $ Banach space . Let $T \in B(X)$ ie. $T$ is a continuous linear map from $X$ to $X$ . define $$P(T)=\sum_{k=o}^{n}a_kT^k$$ with $T^0=id_X$ (identity) . The claim is…
Theorem
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The intuition of the dual space?

The dual space of X is defined to be the space of all linear and continuous functionals that map X to R. But, What exactly is a dual space intuitively? In my current self-guided understanding, I think of a space of function as a set of points( or…
zhongyuan chen
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how to show that $C[0,1]$ is not a Hilbert space with respect to any inner product

Show that the space $C[0, 1]$ of real-valued continuous functions on the unit interval $[0, 1]$ with the sup norm $$ \|f\|=\sup\{|f(x)|:\ x\in[0,1]\} $$ is not a Hilbert space with respect to any inner product . My attempts: as I have to find a…
user476275
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Boundedness of a continuous linear functional on a topological vector space

Suppose: $X$ is topological vector space whose topology is defined by a countable family of separating semi-norms $\|\cdot\|_N$, $N\geq 0$. Suppose $\Lambda:X\to \mathbb{R}$ is a continuous linear functional. Question: Does it follow that there…
Cantor
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On the limits of weakly convergent subsequences

Let $\{ f_n \}$ be a sequence in a Hilbert space $L^2(\mathbb{R}^d)$. We say that this sequence converges weakly to an element $f \in L^2$ if $\langle f_n, g \rangle \to \langle f,g \rangle$ for every $g \in L^2$ (where $\langle \cdot,\cdot \rangle$…
user1736
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Various proofs of Hahn Banach theorem .

I have been seeing various versions of Hahn Banach theorem . There are few confusions that i would like to clarify . 1) In some proof they assume the existence of a sublinear functional and the functional $f$ must satisfy $\le$ condition to the…
Theorem
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When $\|T(b)\|\le M\|b\|$ for each vector from a basis implies that $T$ is bounded?

A linear operator $T\colon X\to Y$ between linear normed spaces is bounded if there exists a constant $M$ such that $$\|Tx\| \le M\|x\|\tag{*}$$ holds for every $x\in X$. Are there some situations when it is sufficient to verify that this condition…
Martin Sleziak
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The dual of the direct sum

Let $X$, $Y$, $Z$ normed spaces If X$\cong Y\oplus Z$ why is $X^*\cong Y^*\oplus Z^*$? where $X^*$ is the dual of $X$. For example ${\ell^\infty}^*\cong\ell^1\oplus\mathrm{Null}\;C_0$ so if we take the double dual we find the…
Student
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Applications of the open mapping theorem for Banach spaces?

Does anybody know of any common/standard/famous practical applications of the open mapping theorem for Banach spaces? Textbooks describe the theorem as a "cornerstone of functional analysis", and yet I have never come across a practical problem that…
Kenny Wong
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$TT^*=T^2$, show that $T$ is self-adjoint

Let $V$ be an inner product space, finitely generated over $\mathbb{C}$, $T\in \operatorname{End}(V)$ that satisfies $TT^*=T^2$, show that $T$ is self-adjoint. I know that $TT^*$ is positive so has positive square-root, thus the square root is…
user145722
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Graphical explanation of Riesz's lemma

Does there exist an intuitive graphical explanation of Riesz's lemma?
Konstantin
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Is $X^*$ complete with weak*-topology

Suppose $X$ is a topological vector space, $X^*$ is its topological dual space. Let the topology of $X^*$ is weak*-topology, Is $X^*$ complete? Suppose $f_s$ is a Cauchy net in $X^*$, it is easy to see that $f=\lim f_s$ exists. We can prove that…
89085731
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Strong convergence does not imply operator norm convergence

How do I construct a sequence of bounded linear transformations that converge strongly to the zero operator, but do not converge to the zero operator in the operator norm? Something strange must happen for certain elements of the Hilbert space, but…
anegligibleperson
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Video lectures on Functional Analysis

I am looking for excellent VIDEO lectures on functional analysis. They should be (1) in English (2) the video quality and voice is good (3) the lecture should not be presented in boring style I am very thankful for your suggestions
Abdulrahman Obaid
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Bounded sets in Frechet spaces

How can I show that, if a Frechet space is not normable then, there is no countable base of bounded sets. A collection $\Gamma$ of subsets of X is called a base for bounded sets, if for any bounded set C there is a $B_{0}$ such that C is contained…
user29253
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