> [!Note] Definition (Finite Dimensional) > A [[Vector Space|vector space ]] $V$ is *finite dimensional* iff it has a finite [[Span of Subset of Vector Space|spanning set]]. > [!Example] Examples > For $\mathbb{R}^{n}$ where $n\in \mathbb{N}$, the standard basis $\underline{e}_{1},\dots,\underline{e}_{n}$ provides a finite spanning set. > > For $\mathbb{R}[x]_{\leq d}$ where $d\in \mathbb{N}$, the monomials $1,x,x^{2}, \dots,x^{d}$ provide a finite spanning set. # Properties Any finite dimensional vector space $V$ has a [[Basis of Vector Space#^cbe5c8|basis]] by [[Basis of Vector Space#^df1788|sifting lemma]]. Also any two bases of $V$ have the same size known as *[[Basis of Vector Space#^b1e1c3|dimension]]* of $V.$ Note [[Dimension of Sum of Finite Dimensional Vector Subspaces (Dimension Formula)]]. Note [[Basis of Vector Space#^61f832|subspaces]] of $V$ must also be finite dimensional with dimension less than or equal to that of $V$. Moreover any [[Linear Independence#^fbfe78|linearly independent subset]] of $V$ has at most $d$ elements where $d$ is the dimension of $V.$ On the other hand, if a vector space is not finite dimensional, then [[Linear Independence#^d0db64|has infinitely many linearly dependent elements]]. Note that the [[Direct sum of vector spaces|direct sum]] of two FDVS is also a FDVS. An FDVS over $\mathbb{R}$ equipped with an inner product is known as a [[Euclidean Space|Euclidean space]].