# Is the set of continuous functions a Banach space?

## Is the set of continuous functions a Banach space?

More generally, the space C(K) of continuous functions on a compact metric space K equipped with the sup-norm is a Banach space. Then Ck([a, b]) is a Banach space with respect to the Ck-norm. Convergence with respect to the Ck-norm is uniform convergence of functions and their first k deriva- tives.

### What is Holder exponent?

The number α is called the exponent of the Hölder condition. A function on an interval satisfying the condition with α > 1 is constant. If α = 1, then the function satisfies a Lipschitz condition. For any α > 0, the condition implies the function is uniformly continuous. The condition is named after Otto Hölder.

#### What is Equicontinuous family function?

In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions.

Is every Banach space a Hilbert space?

Hilbert spaces with their norm given by the inner product are examples of Banach spaces. While a Hilbert space is always a Banach space, the converse need not hold. Therefore, it is possible for a Banach space not to have a norm given by an inner product.

Which of the following is not a Banach space?

The collection of all continuous complex functions on R whose support is compact is denoted by Cc(R). Then the space (Cc(R),‖⋅‖u) is not a Banach space.

## How do you prove that a function is equicontinuous?

Definition 1. A sequence of functions (fn : U → R) is called equicontinuous if for all ϵ > 0 and all x ∈ U there is a δ > 0 such that for all n ∈ N and all y ∈ U if |x − y| < δ then |fn(x) − fn(y)| < ϵ. Basically, this definition says that we may allow δ to depend on x, but δ cannot depend on n.

### How do you prove Equicontinuity?

Lemma 1 Let I be an interval in R and let fn : I → R for n ∈ N. Assume that f is differentiable at all interior points of I (if I is open, that’s all points of I). If there exists M ≥ 0 such that |fn(x)| ≤ M for all x ∈ I0, n ∈ N, then (fn) is equicontinuous. Proof.

#### What is the difference between Hilbert space and Banach space?

A Banach space is normed space that is complete. A Hilbert space is an inner product space that is complete. Since every inner product space is a normed space, a Hilbert space is a complete normed space, i.e. a Banach space.

What is relative compaction used for?

Relative Compaction Test Set is used for laboratory determination of the maximum wet density of soils and aggregates by the California 216 Impact method. Relative compaction is the ratio of in-place wet density to test the maximum wet density of the same soil or aggregate.

What is locally compact?

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood.

## Does every bounded sequence has convergent subsequence?

Theorem. Every bounded sequence of real numbers has a convergent subsequence.

### Can a bounded sequence have a unbounded subsequence?

Yes, an unbounded sequence can have a convergent subsequence. As Weierstrass theorem implies that a bounded sequence always has a convergent subsequence, but it does not stop us from assuming that there can be some cases where unbounded sequence can also lead to some convergent subsequence.

#### Which Banach spaces are not Hilbert spaces?

Banach spaces that are not a Hilbert space are, among many others, ⁿ ⁿ L p ( R ⁿ, d ⁿ x) for p ∈ [ 1, ∞), p ≠ 2. Linear functionals on such spaces can be written as an integral similar to the Hilbert space inner product but in general the functional cannot be associated with an element of the space itself.

What is a Hölder continuous function?

Functions which are locally integrable and whose integrals satisfy an appropriate growth condition are also Hölder continuous. For example, if we let then u is Hölder continuous with exponent α. Functions whose oscillation decay at a fixed rate with respect to distance are Hölder continuous with an exponent that is determined by the rate of decay.

What is a Hölder continuous vector space?

This is a locally convex topological vector space. If the Hölder coefficient is finite, then the function f is said to be (uniformly) Hölder continuous with exponent α in Ω. In this case, the Hölder coefficient serves as a seminorm.

## Are there closed additive subgroups of the Hilbert space?

There are closed additive subgroups of H, not linear subspaces, connected by 1/2–Hölder continuous arcs. An example is the additive subgroup L2 ( R, Z) of the Hilbert space L2 ( R, R ).