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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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$X$ and $Y$ Banach Spaces, $T \in B(X,Y)$, $Y = \operatorname{im}T \oplus M$, for $M \subseteq Y$, then $\operatorname{im}T$ is closed in $Y$

Let $X$ and $Y$ be Banach spaces. If $T \in B(X,Y)$, and $Y = \operatorname{im}T \oplus M$ for some closed linear subspace $M$ of $Y$, then $\operatorname{im}(T)$ is closed in $Y$. I am unsure if this statement is true or not. Nonetheless, I am…
Dragonite
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Is the property of being an inner product space a topological notion?

Let $(E,\lVert\cdot\rVert)$ denote a normed vector space. Recall that an inner product space $E$ is a NVS with an additional gadget, namely an inner product that induces the norm. But, a NVS space $E$ is an IPS if and only if the norm satisfies the…
Dominic Wynter
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Show that $l^2$ is the only $l^p$ space which norm is induced by the inner product

I want to use the theorem of Jordan and von Neumann which states that norm is induced by inner product if and only if the parallelogram law is true Let $\Vert x \Vert_p$ be the norm in $l^p, 1\le p < \infty$. In parallelogram law we have, $\Vert…
janusz
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Banach limit, Hahn-Banach theorem

Let $l_{\mathbb{R}}^{\infty}$ be the linear space of all real-valued bounded sequences over $\mathbb{R}$ for $x=(x_n)_n$ define $$ p(x) = \lim_{n \to \infty} \sup \frac{x_1 + \dots x_n}{n} \\ W = \{x \in l_{\mathbb{R}}^{\infty} : \lim_{n\to \infty}…
Olba12
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How to use open mapping theorem in this case?

If I have $X,Y$ Banach spaces and $T : X \to Y$ a linear and surjective map then $T$ is a open map. I want to prove that if $(y_n)$ is a bounded sequence on $Y$ then there exists a bounded sequence $(x_n)$ on $X$ such that $T(x_n) = y_n$. I want…
L.F. Cavenaghi
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On Mazur-Ulam theorem

From classical theorem of Mazur-Ulam it follows that if $(X,\|\cdot\|)$ Banach space and $T\colon X\to X$ is surjective isometry (i.e. $\|T(x)-T(y)\|=\|x-y\|$ for all $x,y\in X$) with $T(0)=0$ then $T$ is linear. My question is what if we instead of…
golomorfMath
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Counterexample in uniform bounded principle

I would like to give counterexamples to show that the uniform boundedness principle fails if one relaxes the assumptions in any of the following ways: The given space is merely a normed vector space rather than a Banach space (i.e. completeness is…
blindman
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dense subspace of $ c_0( \mathbb N)$

Prove that $$Y= \left\{ x=(x_n)_{n \in\mathbb{N}} \in c_{0}(\mathbb N )~ \Bigg | ~\sum_{n=1}^{\infty} x_n = 0 \right\}$$ is a dense linear subspace of $ c_0( \mathbb N)$. where $ \displaystyle{c_0( \mathbb N) = \left\{ x=(x_n)_{n \in\mathbb{N}}…
passenger
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Space of linear operators complete, but target space not.

Let $X$, $Y$ be normed spaces and denote by $L(X,Y)$ the space of bounded linear operators from $X$ to $Y$. We know that $L(X,Y)$ is complete if $Y$ is. Could you provide me with an example where $L(X,Y)$ is complete, but $Y$ is not?
Severin Schraven
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Does the form domain of the Friedrichs extension of an unbounded operator compactly embed?

Say we have a Hilbert space $H$ and a positive symmetric operator $T$ with domain $D$. Define a norm $\|u\|_T = \langle Tu, u\rangle$ for $u\in D$ and take the completion of $D$ with respect to this norm to obtain a new Hilbert space $V$. Part of…
Neal
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$l^p$ not norm, $p<1$

please I tried to find counterexamples to see that $l^p$ is not norm with $p<1$ in the triangle inequality but I have problems with convergence when I choose some successions. Thanks.
mathreda
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A compact operator is bounded

I'd like to show that a compact linear operator $T: X \to Y$ between normed spaces is bounded. Can you tell me if this is right? If $T$ is compact, then the closure of the image of $B(0,1)$ and hence of $B(0,n)$ is compact. Then $\|T\| = \sup_{\|x\|…
Rudy the Reindeer
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square root commutes with multiplication for positive elements in a $C^*$ algebra?

Let $A$ be a unital $C^*$ algebra. If $z\in A$ is invertible, then so is $z^*$ and $z^*z$ and, furthermore, $z^*z$ is positive, so we can define using the functional calculus $|z|=\sqrt{z^*z}$. My book then claims that $|z|$ is invertible with…
Jake
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Proof that $c_0$ is separable (with respect to the $l_\infty$ norm)

I'm looking at the proof that $c_0$ is separable, but I don't understand the proof.In the proof below, it first shows that $S$ is separable, where $S$ is: Next, it shows that $S$ is dense in $c_0$. However, $S$ is not a countable set, so how does…
nomadicmathematician
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Do projections preserve closed subspaces

Let $H$ be a Hilbert space and let $\pi \colon H \to H$ be an orthogonal projection. Let $E \subset H$ be a closed subspace of $H$. My question: Is there any hope that one can conclude that $\pi(E)$ is closed?
Sebastian
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