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A normed vector space is a pair (V, ‖·‖ ) where V is a vector space and ‖·‖ a norm on V.
We often omit p or ‖·‖ and just write V for a space if it is clear from the context what (semi) norm we are using.
In a more general sense, a vector norm can be taken to be any real-valued vector that satisfies these three properties. The properties 1. and 2. together imply that if and only if x = 0.
A useful variation of the triangle inequality is for any vectors x and y.
This also shows that a vector norm is a continuous function.
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Example of linear functional in dual space?
dual space W* of W can naturally identified with linear functionals
What is a linear square meter for concrete?
The question is nonsense. A linear square metre is like a square circle! Linear refers to length or distance in 1-dimensional space while a square metre is a measure of area in 2-dimensional space.
How much is 2900 square yards in linear feet?
Square yards are a measure of area, i.e. 2-dimensional space, and linear feet are a measure of...well, lines, i.e. 1-dimensional space. One cannot be computed into the other.
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There can be linear equations with 1, 2, ... variables. Each of these is different since an equation with n variables belongs to n-dimensional space.
Who discovered linear equations?
View all Sir William Rowan Hamilton invented the linear equation in 1843.
If T is a linear operator on an inner product space is the norm of Tx equals norm of x if and only if inner product of Tx and Ty equals inner product of x and y?
no -- consider linear map sending entire source space to zero of target space
What is a norm in mathematics?
In linear algebra the norm is the function that assigns a positive length or size to a number. So for example the norm of negative six is six. It is usually denoted with double vertical lines x.
Example of linear functional in dual space?
dual space W* of W can naturally identified with linear functionals
Prove that Hilbert Space is a Metric Space?
The question doesn't make sense, or alternatively it is true by definition. A Hilbert Space is a complete inner product space - complete in the metric induced by the norm defined by the inner product over the space. In other words an inner product space is a vector space with an inner product defined on it. An inner product then defines a norm on the space, and every norm on a space induces a metric. A Hilbert Space is thus also a complete metric space, simply where the metric is induced by the inner product.
What is quasi banach spaces?
It is a vector space with a quasi norm instead of a norm. A quasi norm is a variation of a norm which follows all the norm axioms except for the triangle inequality where we have x+y< or = K(x+y)for some K>1
How can we compute norm in mathematics?
It depends on what space your in. If its the supremum norm on a function space then just look for the max of the function. If its the euclidean norm then just takes squares, add, take the square root. Whats more interesting is that its often very hard to compute norms. For instance, even computing the norm of a 2x2 matrix is no easy problem if the matrix isn't diagonalizable. Computing the norm of a given operator on a infinite dimensional Hilbert space is very hard indeed...
What is norm on vector space?
It tells us how to measure the length of the vectors.
How are linear inequalities different from linear equations?
A linear equation represents a line. A linear inequality represents part of the space on one side (or the other) of the line defined by the corresponding equation.
What is a linear square meter for concrete?
The question is nonsense. A linear square metre is like a square circle! Linear refers to length or distance in 1-dimensional space while a square metre is a measure of area in 2-dimensional space.
What of these is contained in all artwork Linear perspective and hue Positive space and negative space Depth and color?
Positive space and negative space
How many linear feet in a space 30 feet x 32 feet?
A linear foot has no width, so no matter how many you have of them, you will never cover a 32 x 30 space.
What is null space of linear transformation?
The null space describes what gets sent to 0 during the transformation. Also known as the kernel of the transformation. That is, for a linear transformation T, the null space is the set of all x such that T(x) = 0.
